11.6: Solving Systems with Gaussian Elimination
- Write the augmented matrix of a system of equations.
- Write the system of equations from an augmented matrix.
- Perform row operations on a matrix.
- Solve a system of linear equations using matrices.
Carl Friedrich Gauss lived during the late \(18^{th}\) century and early \(19^{th}\) century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables . In this section, we will revisit this technique for solving systems, this time using matrices.
Writing the Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix .
For example, consider the following \(2 × 2\) system of equations.
\[\begin{align*} 3x+4y&= 7\\ 4x-2y&= 5 \end{align*}\]
We can write this system as an augmented matrix:
\(\left[ \begin{array}{cc|c} 3&4&7\\4&-2&5\end{array} \right]\)
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.
\(\begin{bmatrix}3&4\\4&−2\end{bmatrix}\)
A three-by-three system of equations such as
\[\begin{align*} 3x-y-z&= 0\\ x+y&= 5\\ 2x-3z&= 2 \end{align*}\]
has a coefficient matrix
\(\begin{bmatrix}3&−1&−1\\1&1&0\\2&0&−3\end{bmatrix}\)
and is represented by the augmented matrix
\(\left[ \begin{array}{ccc|c}3&−1&−1&0\\1&1&0&5\\2&0&−3&2\end{array} \right]\)
Notice that the matrix is written so that the variables line up in their own columns: \(x\)-terms go in the first column, \(y\)-terms in the second column, and \(z\)-terms in the third column. It is very important that each equation is written in standard form \(ax+by+cz=d\) so that the variables line up. When there is a missing variable term in an equation, the coefficient is \(0\).
- Write the coefficients of the \(x\)-terms as the numbers down the first column.
- Write the coefficients of the \(y\)-terms as the numbers down the second column.
- If there are \(z\)-terms, write the coefficients as the numbers down the third column.
- Draw a vertical line and write the constants to the right of the line.
Write the augmented matrix for the given system of equations.
\[\begin{align*} x+2y-z&= 3\\ 2x-y+2z&= 6\\ x-3y+3z&= 4 \end{align*}\]
Solution
The augmented matrix displays the coefficients of the variables, and an additional column for the constants.
\(\left[ \begin{array}{ccc|c}1&2&−1&3\\2&−1&2&6\\1&−3&3&4\end{array} \right]\)
Write the augmented matrix of the given system of equations.
\[\begin{align*} 4x-3y&= 11\\ 3x+2y&= 4 \end{align*}\]
- Answer
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\(\left[ \begin{array}{cc|c} 4&−3&11\\3&2&4\end{array} \right]\)
Writing a System of Equations from an Augmented Matrix
We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.
Find the system of equations from the augmented matrix.
\(\left[ \begin{array}{ccc|c}1&−3&−5&-2\\2&−5&−4&5\\−3&5&4&6 \end{array} \right]\)
Solution
When the columns represent the variables \(x\), \(y\), and \(z\),
\[\left[ \begin{array}{ccc|c}1&-3&-5&-2\\2&-5&-4&5\\-3&5&4&6 \end{array} \right] \rightarrow \begin{align*} x-3y-5z&= -2\\ 2x-5y-4z&= 5\\ -3x+5y+4z&= 6 \end{align*}\]
Write the system of equations from the augmented matrix.
\(\left[ \begin{array}{ccc|c}1&−1& 1&5\\2&−1&3&1\\0&1&1&-9\end{array}\right]\)
- Answer
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\(\begin{align*} x-y+z&= 5\\ 2x-y+3z&= 1\\ y+z&= -9 \end{align*}\)
Performing Row Operations on a Matrix
Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.
Row-echelon form \(\begin{bmatrix}1&a&b\\0&1&d\\0&0&1\end{bmatrix}\)
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
- In any nonzero row, the first nonzero number is a \(1\). It is called a leading \(1\).
- Any all-zero rows are placed at the bottom on the matrix.
- Any leading \(1\) is below and to the right of a previous leading \(1\).
- Any column containing a leading \(1\) has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
- Interchange rows. (Notation: \(R_i ↔ R_j\))
- Multiply a row by a constant. (Notation: \(cR_i\))
- Add the product of a row multiplied by a constant to another row. (Notation: \(R_i+cR_j\))
Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a \(1\) as the first entry so that row \(1\) can be used to convert the remaining rows.
The Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix \(A\) with the number \(1\) as the entry down the main diagonal and have all zeros below.
\(A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\xrightarrow{After\space Gaussian\space elimination} A=\begin{bmatrix}1&b_{12}& b_{13}\\0&1&b_{23}\\0&0&1\end{bmatrix}\)
The first step of the Gaussian strategy includes obtaining a \(1\) as the first entry, so that row \(1\) may be used to alter the rows below.
- The first equation should have a leading coefficient of \(1\). Interchange rows or multiply by a constant, if necessary.
- Use row operations to obtain zeros down the first column below the first entry of \(1\).
- Use row operations to obtain a \(1\) in row 2, column 2.
- Use row operations to obtain zeros down column 2, below the entry of 1.
- Use row operations to obtain a \(1\) in row 3, column 3.
- Continue this process for all rows until there is a \(1 in every entry down the main diagonal and there are only zeros below.
- If any rows contain all zeros, place them at the bottom.
Solve the given system by Gaussian elimination.
\[\begin{align*} 2x+3y&= 6\\ x-y&= \dfrac{1}{2} \end{align*}\]
Solution
First, we write this as an augmented matrix.
\(\left[ \begin{array}{cc|c} 2&3&6\\1&−1&12\end{array} \right]\)
We want a \(1\) in row 1, column 1. This can be accomplished by interchanging row 1 and row 2.
\(R_1\leftrightarrow R_2\rightarrow \left[ \begin{array}{cc|c} 1&−1&12\\2&3&6\end{array} \right]\)
We now have a \(1\) as the first entry in row 1, column 1. Now let’s obtain a \(0\) in row 2, column 1. This can be accomplished by multiplying row 1 by \(−2\), and then adding the result to row 2.
\(-2R_1+R_2=R_2\rightarrow \left[ \begin{array}{cc|c} 1&−1&12\\0&5&5\end{array} \right]\)
We only have one more step, to multiply row 2 by \(\dfrac{1}{5}\).
\(\dfrac{1}{5}R_2=R_2\rightarrow \left[ \begin{array}{cc|c} 1&−1&12\\0&1&1\end{array} \right]\)
Use back-substitution. The second row of the matrix represents \(y=1\). Back-substitute \(y=1\) into the first equation.
\[\begin{align*} x-(1)&= \dfrac{1}{2}\\ x&= \dfrac{3}{2} \end{align*}\]
The solution is the point \(\left(\dfrac{3}{2},1\right)\).
Solve the given system by Gaussian elimination.
\[\begin{align*} 4x+3y&= 11\\ x-3y&= -1 \end{align*}\]
- Answer
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\((2, 1)\)
Use Gaussian elimination to solve the given \(2 × 2\) system of equations .
\[\begin{align*} 2x+y&= 1\\ 4x+2y&= 6 \end{align*}\]
Solution
Write the system as an augmented matrix .
\(\left[ \begin{array}{cc|c} 2&1&1\\4&2&6\end{array} \right]\)
Obtain a \(1\) in row 1, column 1. This can be accomplished by multiplying the first row by \(\dfrac{1}{2}\).
\(\dfrac{1}{2} R_1=R_1\rightarrow \left[ \begin{array}{cc|c} 1&\dfrac{1}{2}&\dfrac{1}{2}\\4&2&6\end{array} \right]\)
Next, we want a \(0\) in row 2, column 1. Multiply row 1 by \(−4\) and add row 1 to row 2.
\(-4R_1+R_2=R_2\rightarrow \left[ \begin{array}{cc|c} 1&\dfrac{1}{2}&\dfrac{1}{2}\\0&0&4\end{array} \right]\)
The second row represents the equation \(0=4\). Therefore, the system is inconsistent and has no solution.
Solve the system of equations.
\[\begin{align*} 3x+4y&= 12\\ 6x+8y&= 24 \end{align*}\]
Solution
Perform row operations on the augmented matrix to try and achieve row-echelon form .
\(A=\left[ \begin{array}{cc|c} 3&4&12\\6&8&24\end{array} \right]\)
\(-\dfrac{1}{2}R_2+R_1=R_1\rightarrow \left[ \begin{array}{cc|c} 0&0&0\\6&8&24\end{array} \right]\)
\(R_1\leftrightarrow R_2=\left[ \begin{array}{cc|c} 6&8&24\\0&0&0\end{array} \right]\)
The matrix ends up with all zeros in the last row: \(0y=0\). Thus, there are an infinite number of solutions and the system is classified as dependent. To find the generic solution, return to one of the original equations and solve for \(y\).
\[\begin{align*} 3x+4y&= 12\\ 4y&= 12-3x\\ y&= 3-\dfrac{3}{4}x \end{align*}\]
So the solution to this system is \(\left(x,3−\dfrac{3}{4}x\right)\).
Perform row operations on the given matrix to obtain row-echelon form.
\(\left[ \begin{array}{ccc|c} 1&-3&4&3\\2&-5&6&6\\-3&3&4&6\end{array} \right]\)
Solution
The first row already has a \(1\) in row 1, column 1. The next step is to multiply row 1 by \(−2\) and add it to row 2. Then replace row 2 with the result.
\(-2R_1+R_2=R_2 \left[ \begin{array}{ccc|c} 1&-3&4&3\\0&1&-2&0\\-3&3&4&6\end{array} \right]\)
Next, obtain a zero in row 3, column 1.
\(3R_1+R_3=R_3 \left[ \begin{array}{ccc|c} 1&-3&4&3\\0&1&-2&0\\0&-6&16&15\end{array} \right]\)
Next, obtain a zero in row 3, column 2.
\(6R_2+R_3=R_3 \left[ \begin{array}{ccc|c} 1&-3&4&3\\0&1&-2&0\\0&0&4&15\end{array} \right]\)
The last step is to obtain a 1 in row 3, column 3.
\(\dfrac{1}{4}R_3=R_3 \left[ \begin{array}{ccc|c} 1&-3&4&3\\0&1&-2&0\\0&0&1&\dfrac{15}{4}\end{array} \right]\)
Write the system of equations in row-echelon form.
\[\begin{align*} x−2y+3z &= 9 \\ −x+3y &= −4 \\ 2x−5y+5z &= 17 \end{align*}\]
- Answer
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\(\left[ \begin{array}{ccc|c} 1&-\dfrac{5}{2}&\dfrac{5}{2}&\dfrac{17}{2}\\0&1&5&9\\0&0&1&2\end{array} \right]\)
Solving a System of Linear Equations Using Matrices
We have seen how to write a system of equations with an augmented matrix , and then how to use row operations and back-substitution to obtain row-echelon form. Now, we will take row-echelon form a step farther to solve a \(3\) by \(3\) system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.
Solve the system of linear equations using matrices.
\[\begin{align*} x-y+z&= 8\\ 2x+3y-z&= -2\\ 3x-2y-9z&= 9 \end{align*}\]
Solution
First, we write the augmented matrix.
\(\left[ \begin{array}{ccc|c} 1&-1&1&8\\2&3&-1&-2\\3&-2&-9&9\end{array} \right]\)
Next, we perform row operations to obtain row-echelon form.
\(−2R_1+R_2=R_2\rightarrow \left[ \begin{array}{ccc|c} 1&-1&1&8\\0&5&-3&-18\\3&-2&-9&9\end{array} \right]\)
\(−3R_1+R_3=R_3\rightarrow \left[ \begin{array}{ccc|c} 1&-1&1&8\\0&5&-3&-18\\0&1&-12&-15\end{array} \right]\)
The easiest way to obtain a \(1\) in row 2 of column 1 is to interchange \(R_2\) and \(R_3\).
\(Interchange\space R_2\space and\space R_3\rightarrow\left[ \begin{array}{ccc|c} 1&-1&1&8\\0&1&-12&-15\\0&5&-3&-18\end{array} \right]\)
Then
\(−5R_2+R_3=R_3\rightarrow\left[ \begin{array}{ccc|c} 1&-1&1&8\\0&1&-12&-15\\0&0&57&57\end{array} \right]\)
\(−\dfrac{1}{57}R_3=R_3\rightarrow\left[ \begin{array}{ccc|c} 1&-1&1&8\\0&1&-12&-15\\0&0&1&1\end{array} \right]\)
The last matrix represents the equivalent system.
\[\begin{align*} x−y+z &= 8 \\ y−12z &= −15 \\ z &= 1 \end{align*}\]
Using back-substitution, we obtain the solution as \((4,−3,1)\).
Solve the following system of linear equations using matrices.
\[\begin{align*} −x−2y+z &= −1 \\ 2x+3y &= 2 \\ y−2z &= 0 \end{align*}\]
Solution
Write the augmented matrix.
\(\left[ \begin{array}{ccc|c} -1&-2&1&-1\\2&3&0&2\\0&1&-2&0\end{array} \right]\)
First, multiply row 1 by \(−1\) to get a \(1\) in row 1, column 1. Then, perform row operations to obtain row-echelon form.
\(-R_1\rightarrow \left[ \begin{array}{ccc|c} 1&2&-1&1\\2&3&0&2\\0&1&-2&0\end{array} \right]\)
\(R_2\leftrightarrow R_3\rightarrow \left[ \begin{array}{ccc|c} 1&2&-1&1\\0&1&-2&0\\2&3&0&2\end{array} \right]\)
\(−2R_1+R_3=R_3\rightarrow \left[ \begin{array}{ccc|c} 1&2&-1&1\\0&1&-2&0\\0&-1&2&0\end{array} \right]\)
\(R_2+R_3=R_3\rightarrow \left[ \begin{array}{ccc|c} 1&2&-1&1\\0&1&-2&0\\0&0&0&0\end{array} \right]\)
The last matrix represents the following system.
\[\begin{align*} x+2y−z &= 1 \\ y−2z &= 0 \\ 0 &= 0 \end{align*}\]
We see by the identity \(0=0\) that this is a dependent system with an infinite number of solutions. We then find the generic solution. By solving the second equation for \(y\) and substituting it into the first equation we can solve for \(z\) in terms of \(x\).
\[\begin{align*} x+2y−z &= 1 \\ y &= 2z \\ x+2(2z)−z &= 1 \\ x+3z &= 1 \\ z &=\dfrac{1−x}{3} \end{align*}\]
Now we substitute the expression for \(z\) into the second equation to solve for \(y\) in terms of \(x\).
\[\begin{align*} y−2z &= 0 \\ z &= \dfrac{1−x}{3} \\ y−2\left(\dfrac{1−x}{3}\right) &= 0 \\ y &= \dfrac{2−2x}{3} \end{align*}\]
The generic solution is \(\left(x,\dfrac{2−2x}{3},\dfrac{1−x}{3}\right)\).
Solve the system using matrices.
\[\begin{align*} x+4y-z&= 4\\ 2x+5y+8z&= 1\\ 5x+3y-3z&= 1 \end{align*}\]
- Answer
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\((1,1,1)\)
Yes, a system of linear equations of any size can be solved by Gaussian elimination.
- Save the augmented matrix as a matrix variable \([A], [B], [C], ….\)
- Use the ref( function in the calculator, calling up each matrix variable as needed.
Solve the system of equations.
\[\begin{align*} 5x+3y+9z&= -1\\ -2x+3y-z&= -2\\ -x-4y+5z&= 1 \end{align*}\]
Solution
Write the augmented matrix for the system of equations.
\(\left[ \begin{array}{ccc|c} 5&3&9&-1\\-2&3&-1&-2\\-1&-4&5&1\end{array} \right]\)
On the matrix page of the calculator, enter the augmented matrix above as the matrix variable \([A]\).
\([A]=\left[ \begin{array}{ccc|c} 5&3&9&-1\\-2&3&-1&-2\\-1&-4&5&1\end{array} \right]\)
Use the ref( function in the calculator, calling up the matrix variable \([A]\).
ref([A])
Evaluate
\[\begin{array}{cc} {\left[ \begin{array}{ccc|c} 1&\dfrac{3}{5}&\dfrac{9}{5}&\dfrac{1}{5}\\0&1&\dfrac{13}{21}&-\dfrac{4}{7}\\0&0&1&-\dfrac{24}{187}\end{array} \right] \rightarrow} & {\begin{align*} x+\dfrac{3}{5}y+\dfrac{9}{5}z &= -\dfrac{1}{5} \\ y+\dfrac{13}{21}z &= -\dfrac{4}{7} \\ z &= -\dfrac{24}{187} \end{align*}} \end{array}\]
Using back-substitution, the solution is \(\left(\dfrac{61}{187},−\dfrac{92}{187},−\dfrac{24}{187}\right)\).
Carolyn invests a total of \($12,000\) in two municipal bonds, one paying \(10.5%\) interest and the other paying \(12%\) interest. The annual interest earned on the two investments last year was \($1,335\). How much was invested at each rate?
Solution
We have a system of two equations in two variables. Let \(x=\) the amount invested at \(10.5%\) interest, and \(y=\) the amount invested at \(12%\) interest.
\[\begin{align*} x+y&= 12,000\\ 0.105x+0.12y&= 1,335 \end{align*}\]
As a matrix, we have
\(\left[ \begin{array}{cc|c} 1&1&12,000\\0.105&0.12&1,335\end{array} \right]\)
Multiply row 1 by \(−0.105\) and add the result to row 2.
\(\left[ \begin{array}{cc|c} 1&1&12,000\\0&0.015&75\end{array} \right]\)
Then,
\[\begin{align*} 0.015y &= 75 \\ y &= 5,000 \end{align*}\]
So \(12,000−5,000=7,000\).
Thus, \($5,000\) was invested at \(12%\) interest and \($7,000\) at \(10.5%\) interest.
Ava invests a total of \($10,000\) in three accounts, one paying \(5%\) interest, another paying \(8%\) interest, and the third paying \(9%\) interest. The annual interest earned on the three investments last year was \($770\). The amount invested at \(9%\) was twice the amount invested at \(5%\). How much was invested at each rate?
Solution
We have a system of three equations in three variables. Let \(x\) be the amount invested at \(5%\) interest, let \(y\) be the amount invested at \(8%\) interest, and let \(z\) be the amount invested at \(9%\) interest. Thus,
\[\begin{align*} x+y+z &= 10,000 \\ 0.05x+0.08y+0.09z &= 770 \\ 2x−z &= 0 \end{align*}\]
As a matrix, we have
\(\left[ \begin{array}{ccc|c} 1&1&1&10,000\\0.05&0.08&0.09&770\\2&0&-1&0\end{array} \right]\)
Now, we perform Gaussian elimination to achieve row-echelon form.
\(−0.05R_1+R_2=R_2\rightarrow \left[ \begin{array}{ccc|c} 1&1&1&10,000\\0&0.03&0.04&270\\2&0&-1&0\end{array} \right]\)
\(−2R_1+R_3=R_3\rightarrow \left[ \begin{array}{ccc|c} 1&1&1&10,000\\0&0.03&0.04&270\\0&-2&-3&-20,000\end{array} \right]\)
\(\dfrac{1}{0.03}R_2=R_2\rightarrow \left[ \begin{array}{ccc|c} 1&1&1&10,000\\0&1&\dfrac{4}{3}&9,000\\0&-2&-3&-20,000\end{array} \right]\)
\(2R_2+R_3=R_3\rightarrow \left[ \begin{array}{ccc|c} 1&1&1&10,000\\0&1&\dfrac{4}{3}&9,000\\0&0&-\dfrac{1}{3}&-2,000\end{array} \right]\)
The third row tells us \(−\dfrac{1}{3}z=−2,000\); thus \(z=6,000\).
The second row tells us \(y+\dfrac{4}{3}z=9,000\). Substituting \(z=6,000\),we get
\[\begin{align*} y+\dfrac{4}{3}(6,000) &= 9,000 \\ y+8,000 &= 9,000 \\ y &= 1,000 \end{align*}\]
The first row tells us \(x+y+z=10,000\). Substituting \(y=1,000\) and \(z=6,000\),we get
\[\begin{align*} x+1,000+6,000 &= 10,000 \\ x &= 3,000 \end{align*}\]
The answer is \($3,000\) invested at \(5%\) interest, \($1,000\) invested at \(8%\), and \($6,000\) invested at \(9%\) interest.
A small shoe company took out a loan of \($1,500,000\) to expand their inventory. Part of the money was borrowed at \(7%\), part was borrowed at \(8%\), and part was borrowed at \(10%\). The amount borrowed at \(10%\) was four times the amount borrowed at \(7%\), and the annual interest on all three loans was \($130,500\). Use matrices to find the amount borrowed at each rate.
- Answer
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\($150,000\) at \(7%\), \($750,000\) at \(8%\), \($600,000\) at \(10%\)
Access these online resources for additional instruction and practice with solving systems of linear equations using Gaussian elimination.
Key Concepts
- An augmented matrix is one that contains the coefficients and constants of a system of equations. See Example \(\PageIndex{1}\).
- A matrix augmented with the constant column can be represented as the original system of equations. See Example \(\PageIndex{2}\).
- Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
- We can use Gaussian elimination to solve a system of equations. See Example \(\PageIndex{3}\), Example \(\PageIndex{4}\), and Example \(\PageIndex{5}\).
- Row operations are performed on matrices to obtain row-echelon form. See Example \(\PageIndex{6}\).
- To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See Example \(\PageIndex{7}\) and Example \(\PageIndex{8}\).
- A calculator can be used to solve systems of equations using matrices. See Example \(\PageIndex{9}\).
- Many real-world problems can be solved using augmented matrices. See Example \(\PageIndex{10}\) and Example \(\PageIndex{11}\).


By: Kerri May
Learning Objectives
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Understand the positives and the negatives for grading on effort.
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Understand the risks for grading on effort
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Decide for themselves what the best option is
Introduction
Envision this scenario: A young boy in elementary school struggles to keep up with his peers. He lags behind in reading, math, and writing. He tries his hardest, but he can never seem to close the gap. His teacher watches him struggle. She sees how hard he works and how much effort he puts into all of his assignments. She wishes that grading was more arbitrary and less competitive- you either pass or you fail. It breaks her heart to watch her students work so hard, only to be disappointed. And sure enough, when report cards come, she watches as his smile fades and apprehension sets in.
Not all children are equal. They learn in different ways. They think differently. Their skills and talents vary. So, then, should they all be graded on the same scale? What kind of scale? And what
is
fair? These are all questions that one must think about when considering the positives and the negatives of grading on effort.
The positives
Have you ever tried your best on an assignment, really gave it your all, only to get a lower grade? Have you ever studied tirelessly for a test, only to bomb it? Sure, we all have. It is inevitable. Children have varying degrees of intelligence and skills. Some are brilliant, others are just average. As teachers, it is our job to try our hardest to never classify students based on intelligence, or any other stereotypical characteristic. But we see it. We see the students that struggle to keep up, and the students who barely try and still succeed with flying colors. And it's not fair.
Therefore, some people believe that students should be graded on effort alone. Not on the quality of their work, but on the effort they put into it. Others believe that a certain percent of the grade should be effort-based. A prime example of an effort-based grading scale is the scale used at the predominantly African American Benedict College of South Carolina. According to the Office of Academic Affairs at Benedict College, the grading scale for freshmen is 60 percent effort, and 40 percent knowledge. Sophomores have a 60 percent knowledge and 40 percent effort (Swinton, D., 2004). The idea of the college's policy, called Success Equals Effort(SEE), is to "provide our students with a strong and immediate incentive to adopt the behaviors and habits that are most likely to bring academic success" (Swinton, D., 2004). They believe that in their unique setting, students are able to receive more feeback than they would in a traditional college setting, and therefore have more room to improve on their skills (Swinton, D., 2004).
In response to an increased amount of negative press, which is discussed below, Benedict College has been forced to defend what some consider questionable practices. With so much opposition to one college, whose grading scale is not based on effort alone, it is easy to see why many public schools might fear a change. Most would not even think of challenging the grading system that has been in practice for so many years. However, many others believe that an effort-based grading scale would be very beneficial for all students.
The Negatives
Fairness and equality for all sound nice, but is it a realistic goal? Many believe that it is not. Michael Covington, a professor at the University of Georgia, strongly believes in assigning a fair letter grading scale. He reasons that the main function of grades is to act as a system of communication between the teacher and the student and between different teachers and educational institutions (Covington, 2004). The teacher assigns the student a grade that he or she earned, based on his or her performance. Everything is clear-cut and straightforward. He believes that students should understand the grading scale, know what is expected of them, and work to achieve that goal to the best of their abilities.
His second principle is very simple: "Grades measure results, not effort" (Covington, 2004). While this may sound harsh, he maintains that in many instances, we do not know how hard our students work. Observing classwork is one thing, but knowing whether students worked hard outside of the classroom is impossible. Furthermore, he adds that "grading on effort can conceal incompetence or, at best, send students into advanced courses for which they are not prepared" (Covington, 2004). Not only that, but students are given degrees based on talent, not on hard work. He reminds us that we would not want a surgeon operating on us that tried his best in college and was graded "on effort" (Covington, 2004).
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Covington's Grading Scale
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Explanation of letter grade
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A
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The student did as well as could reasonably be expected.
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B
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The student’s mastery of the material has noticeable flaws but is well above the minimum standard.
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C
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The student met the minimum requirements for the course.
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D
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The student learned some of the material but did not meet minimum requirements.
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F
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The student learned little or none of the material.
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Walter Williams, a well-known writer with a doctorate in economics from the University of California at Los Angeles, shares a similar view. He provides his opinion regarding the unique grading scale of Benedict College in Columbia, S.C., which grades freshmen and sophomore students on a scale in which effort is weighted greater than or equal to knowledge. He reminds us that "60 percent of a freshman's grade is based on effort and the rest on academic performance" (Williams, 2004). This means that a student could receive an A grade in effort, and an F in knowledge and still pass the class with a C. The reason for this is the SEE Policy, or "Success Equals Effort" policy, advocated by the college.  This is precisely what Williams discusses in article. He believes that a student without the necessary talent that is graded on effort will "fall further and further behind because he hasn't grasped the material from the earlier courses. He'll graduate only if the fraudulent grading continues" (Williams, 2004)). Basically what Williams is trying to say is that students graded on effort, especially at a college level, is completely unrealistic. It does not prepare students for the real world, where knowledge reins supreme. He belives that students attending this college will be in for a rude awakening when they graduate from college and get a real job. Furthermore, Williams believes that degrees from Benedict College may become obsolete, as employers discover the means by which their employees earned their degrees (Williams, 2004).
The Risks
Grades based on effort are nice, in theory. However, even with good intentions we have seen that it produces negative effects. Perhaps in the future, things will change. But for right now, there are too many risks involved. Here are some things that we must consider: What happens to the students who are not challenged? What if they lose interest in school, because it is just too easy? What if students who have special needs, or students who need further instruction are overlooked because they try their best? There are too many factors, in my mind at least, that prevent effort-based grading from being successful.
Exercise \(\PageIndex{1}\)
1. Which is a description of effort-based grading?
A. Students are graded using traditional letter grades A-F.
B. Students are graded using traditional letter grades A-F, with pluses and minuses.
C. Students are graded based on how hard they work, sometimes in combination with their knowledge.
D. Students are graded on knowledge, using a number scale of 1-5.
2. Which is a benefit for effort-based grading?
A. Students that may not be as gifted but try their hardest may receive good grades.
B. Students that are extremely bright are able to use their skills to help other students.
C. Inclusion students will be given opportunities to broaden their knowledge.
D. Students that do not do their work can still receive good grades.
3. Mrs. Smith's elementary school wants to implement a effort-based grading program. She is afraid that her students will suffer from this change in policy. What statement most effectively argues against effort-based grading?
A. Gifted students may not feel challenged, and lose interest in school.
B. Average students will not be able to work to the best of their abilities.
C. Inclusion students will face a disadvantage.
D. Gifted students will become much more advanced than the rest of the class, creating animosity between students.
4. Jimmy works very hard in school. He does all of his homework, studies hard for tests, and listens attentively in school. His grades are excellent. However, Jimmy does not always grasp the material as quickly as the other students, and is usually struggling. He attends a school that employs an effort-based grading scale. Jimmy's deficiencies are not noticed by his teacher. What is the most likely reason?
A. Jimmy hides his confusion and still manages to do well by copying his classmates' work.
B. Jimmy tries his best and receives an A for effort, which is most important in their school.
C. Jimmy's teacher is too busy with paperwork to notice.
D. Jimmy's classmates are all on the same level that he is.
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Answer
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1. C
2. A
3. A
4. B
VII. References
Covington, Michael A. (2004). What should grades mean? Retrieved October 24, 2008, from www.ai.uga.edu/mc/grading.html.
Swinton, David Holmes. (2004). Office of Academic Affairs: Open Letter Regarding The Success Equals Effort (SEE) Policy. Retrieved from: www.benedict.edu/divisions/acadaf/office/policy_n_proc/bc-acad_affairs-see_policy-letter.html.
Swinton, Omari H. (2006). The Effect of Effort Grading on Learning. Power Point Presentation given at Pipeline Conference. Retrieved from
http://www.vanderbilt.edu/AEA/CSMGEP/pipeline/06conference_files/Omari_Swinton_Presentation.pdf
.
Williams, Walter. (2004). The academic halls of stupidity: Success Equals Effort.
Capitalism Magazine.
Retrieved October 25, 2008, from
http://www.capmag.com/article.asp?ID=3967
.


5.3E: Line Integrals (Exercises)
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1. True or False? Line integral \(\displaystyle\int _C f(x,y)\,ds\) is equal to a definite integral if \(C\) is a smooth curve defined on \([a,b]\) and if function \(f\) is continuous on some region that contains curve \(C\).
- Answer
- True
2. True or False? Vector functions \(\vecs r_1=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}, \quad 0≤t≤1,\) and \(\vecs r_2=(1−t)\,\hat{\mathbf i}+(1−t)^2\,\hat{\mathbf j}, \quad 0≤t≤1\), define the same oriented curve.
3. True or False? \(\displaystyle\int _{−C}(P\,dx+Q\,dy)=\int _C(P\,dx−Q\,dy)\)
- Answer
- False
4. True or False? A piecewise smooth curve \(C\) consists of a finite number of smooth curves that are joined together end to end.
5. True or False? If \(C\) is given by \(x(t)=t,\quad y(t)=t, \quad 0≤t≤1\), then \(\displaystyle\int _Cxy\,ds=\int ^1_0t^2\,dt.\)
- Answer
- False
For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.
6. [T] \(\displaystyle\int _C(x+y)\,ds\)
\(C:x=t,y=(1−t),z=0\) from \((0, 1, 0)\) to \((1, 0, 0)\)
7. [T] \(\displaystyle \int _C(x−y)ds\)
\(C:\vecs r(t)=4t\,\hat{\mathbf i}+3t\,\hat{\mathbf j}\) when \(0≤t≤2\)
- Answer
- \(\displaystyle\int _C(x−y)\,ds=10\)
8. [T] \(\displaystyle\int _C(x^2+y^2+z^2)\,ds\)
\(C:\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j}+8t\,\hat{\mathbf k}\) when \(0≤t≤\dfrac{π}{2}\)
9. [T] Evaluate \(\displaystyle\int _Cxy^4\,ds\), where \(C\) is the right half of circle \(x^2+y^2=16\) and is traversed in the clockwise direction.
- Answer
- \(\displaystyle\int _Cxy^4\,ds=\frac{8192}{5}\)
10. [T] Evaluate \(\displaystyle\int _C4x^3ds\), where C is the line segment from \((−2,−1)\) to \((1, 2)\).
For the following exercises, find the work done.
11. Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1,4,2)\) to \((0,5,1)\).
- Answer
- \(W=8\) units of work
12. Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.
13. Find the work done by force field \(\vecs F(x,y,z)=−\dfrac{1}{2}x\,\hat{\mathbf i}−\dfrac{1}{2}y\,\hat{\mathbf j}+\dfrac{1}{4}\,\hat{\mathbf k}\) on a particle as it moves along the helix \(\vecs r(t)=\cos t\,\hat{\mathbf i}+\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k}\) from point \((1,0,0)\) to point \((−1,0,3π)\).
- Answer
- \(W=\frac{3π}{4}\) units of work
14. Find the work done by vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+2x\,\hat{\mathbf j}\) in moving an object along path \(C\), which joins points \((1, 0)\) and \((0, 1)\).
15. Find the work done by force \(\vecs{F}(x,y)=2y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}+(x+y)\,\hat{\mathbf k}\) in moving an object along curve \(\vecs r(t)=\cos(t)\,\hat{\mathbf i}+\sin(t)\,\hat{\mathbf j}+16\,\hat{\mathbf k}\), where \(0≤t≤2π\).
- Answer
- \(W=π\) units of work
16. Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density \(ρ(x,y)=y^2\).
For the following exercises, evaluate the line integrals.
17. Evaluate \(\displaystyle\int_C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=−1\,\hat{\mathbf j}\), and \(C\) is the part of the graph of \(y=\frac{1}{2}x^3−x\) from \((2,2)\) to \((−2,−2)\).
- Answer
- \(\displaystyle\int _C\vecs F·d\vecs{r}=4\) units of work
18. Evaluate \(\displaystyle\int _γ(x^2+y^2+z^2)^{−1}ds\), where \(γ\) is the helix \(x=\cos t,y=\sin t,z=t,\) with \(0≤t≤T.\)
19. Evaluate \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz\) over the line segment from \((1,1,1) \) to \((3,2,0).\)
- Answer
- \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz=−1\)
20. Let \(C\) be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral \(\displaystyle\int _Cy\,ds.\)
21. [T] Use a computer algebra system to evaluate the line integral \(\displaystyle\int _Cy^2\,dx+x\,dy\), where \(C\) is the arc of the parabola \(x=4−y^2\) from \((−5, −3)\) to \((0, 2)\).
- Answer
- \(\displaystyle\int _C(y^2)\,dx+(x)\,dy=\dfrac{245}{6}\)
22. [T] Use a computer algebra system to evaluate the line integral \(\displaystyle\int _C (x+3y^2)\,dy\) over the path \(C\) given by \(x=2t,y=10t,\) where \(0≤t≤1.\)
23. [T] Use a CAS to evaluate line integral \(\displaystyle\int _C xy\,dx+y\,dy\) over path \(C\) given by \(x=2t,y=10t\), where \(0≤t≤1\).
- Answer
- \(\displaystyle\int _Cxy\,dx+y\,dy=\dfrac{190}{3}\)
24. Evaluate line integral \(\displaystyle\int _C(2x−y)\,dx+(x+3y)\,dy\), where \(C\) lies along the \(x\)-axis from \(x=0\) to \(x=5\).
26. [T] Use a CAS to evaluate \(\displaystyle\int _C\dfrac{y}{2x^2−y^2}\,ds\), where \(C\) is defined by the parametric equations \(x=t,y=t\), for \(1≤t≤5.\)
- Answer
- \(\displaystyle\int _C\frac{y}{2x^2−y^2}\,ds=\sqrt{2}\ln 5\)
27. [T] Use a CAS to evaluate \(\displaystyle\int _Cxy\,ds\), where \(C\) is defined by the parametric equations \(x=t^2,y=4t\), for \(0≤t≤1.\)
In the following exercises, find the work done by force field \(\vecs F\) on an object moving along the indicated path.
28. \(\vecs{F}(x,y)=−x \,\hat{\mathbf i}−2y\,\hat{\mathbf j}\)
\(C:y=x^3\) from \((0, 0)\) to \((2, 8)\)
- Answer
- \(W=−66\) units of work
29. \(\vecs{F}(x,y)=2x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)
<\(C\): counterclockwise around the triangle with vertices \((0, 0), (1, 0), \) and \((1, 1)\)
30. \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}−5z\,\hat{\mathbf k}\)
\(C:\vecs r(t)=2\cos t\,\hat{\mathbf i}+2\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k},\; 0≤t≤2π\)
- Answer
- \(W=−10π^2\) units of work
31. Let \(\vecs F\) be vector field \(\vecs{F}(x,y)=(y^2+2xe^y+1)\,\hat{\mathbf i}+(2xy+x^2e^y+2y)\,\hat{\mathbf j}\). Compute the work of integral \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(C\) is the path \(\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j},\quad 0≤t≤\dfrac{π}{2}\).
32. Compute the work done by force \(\vecs F(x,y,z)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}−z\,\hat{\mathbf k}\) along path \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+t^3\,\hat{\mathbf k}\), where \(0≤t≤1\).
- Answer
- \(W=2\) units of work
33. Evaluate \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=\dfrac{1}{x+y}\,\hat{\mathbf i}+\dfrac{1}{x+y}\,\hat{\mathbf j}\) and \(C\) is the segment of the unit circle going counterclockwise from \((1,0)\) to \((0, 1)\).
34. Force \(\vecs F(x,y,z)=zy\,\hat{\mathbf i}+x\,\hat{\mathbf j}+z^2x\,\hat{\mathbf k}\) acts on a particle that travels from the origin to point \((1, 2, 3)\). Calculate the work done if the particle travels:
- along the path \((0,0,0)→(1,0,0)→(1,2,0)→(1,2,3)\) along straight-line segments joining each pair of endpoints;
- along the straight line joining the initial and final points.
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Is the work the same along the two paths?
- Answer
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a. \(W=11\) units of work;
b. \(W=\dfrac{39}{4}=9\frac{3}{4}\) units of work;
c. No
35. Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1, 4, 2)\) to \((0, 5, 1).\)
36. How much work is required to move an object in vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}\) along the upper part of ellipse \(\dfrac{x^2}{4}+y^2=1\) from \((2, 0)\) to \((−2,0)\)?
- Answer
- \(W=2π\) units of work
37. A vector field is given by \(\vecs{F}(x,y)=(2x+3y)\,\hat{\mathbf i}+(3x+2y)\,\hat{\mathbf j}\). Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.
38. Evaluate the line integral of scalar function \(xy\) along parabolic path \(y=x^2\) connecting the origin to point \((1, 1)\).
- Answer
- \(\displaystyle\int _C f\,ds=\dfrac{25\sqrt{5}+1}{120}\)
39. Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y=3x\) from \((0, 0)\) to \((1, 3).\)
40. Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y^2=9x\) from \((0, 0)\) to \((1, 3).\)
- Answer
- \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy=6.15\)
For the following exercises, use a CAS to evaluate the given line integrals.
41. [T] Evaluate \(\vecs F(x,y,z)=x^2z\,\hat{\mathbf i}+6y\,\hat{\mathbf j}+yz^2\,\hat{\mathbf k}\), where \(C\) is represented by \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+\ln t \,\hat{\mathbf k},1≤t≤3\).
42. [T] Evaluate line integral \(\displaystyle\int _γxe^y\,ds\) where, \(γ\) is the arc of curve \(x=e^y\) from \((1,0)\) to \((e,1)\).
- Answer
- \(\displaystyle\int _γxe^y\,ds≈7.157\)
43. [T] Evaluate the integral \(\displaystyle\int _γxy^2\,ds\), where \(γ\) is a triangle with vertices \((0, 1, 2), (1, 0, 3)\), and \((0,−1,0)\).
44. [T] Evaluate line integral \(\displaystyle\int _γ(y^2−xy)\,dx\), where \(γ\) is curve \(y=\ln x\) from \((1, 0)\) toward \((e,1)\).
- Answer
- \(\displaystyle\int _γ(y^2−xy)\,dx≈−1.379\)
45. [T] Evaluate line integral \(\displaystyle\int_γ xy^4\,ds\), where \(γ\) is the right half of circle \(x^2+y^2=16\).
46. [T] Evaluate \(\int C \vecs F⋅d\vecs{r},\int C \vecs F·d\vecs{r},\) where \(\vecs F(x,y,z)=x^2y\,\mathbf{\hat i}+(x−z)\,\mathbf{\hat j}+xyz\,\mathbf{\hat k}\) and
\(C: \vecs r(t)=t\,\mathbf{\hat i}+t^2\,\mathbf{\hat j}+2\,\mathbf{\hat k},0≤t≤1\).
- Answer
- \(\displaystyle\int _C \vecs F⋅d\vecs{r}≈−1.133\) units of work
47. Evaluate \(\displaystyle\int _C \vecs F⋅d\vecs{r}\), where \(\vecs{F}(x,y)=2x\sin y\,\mathbf{\hat i}+(x^2\cos y−3y^2)\,\mathbf{\hat j}\) and
\(C\) is any path from \((−1,0)\) to \((5, 1)\).
48. Find the line integral of \(\vecs F(x,y,z)=12x^2\,\mathbf{\hat i}−5xy\,\mathbf{\hat j}+xz\,\mathbf{\hat k}\) over path \(C\) defined by \(y=x^2, z=x^3\) from point \((0, 0, 0)\) to point \((2, 4, 8)\).
- Answer
- \(\displaystyle\int _C \vecs F⋅d\vecs{r}≈22.857\) units of work
49. Find the line integral of \(\displaystyle\int _C(1+x^2y)\,ds\), where \(C\) is ellipse \(\vecs r(t)=2\cos t\,\mathbf{\hat i}+3\sin t\,\mathbf{\hat j}\) from \(0≤t≤π.\)
For the following exercises, find the flux.
50. Compute the flux of \(\vecs{F}=x^2\,\mathbf{\hat i}+y\,\mathbf{\hat j}\) across a line segment from \((0, 0)\) to \((1, 2).\)
- Answer
- \(\text{flux}=−\frac{1}{3}\)
51. Let \(\vecs{F}=5\,\mathbf{\hat i}\) and let \(C\) be curve \(y=0,\) with \(0≤x≤4\). Find the flux across \(C\).
52. Let \(\vecs{F}=5\,\mathbf{\hat j}\) and let \(C\) be curve \(y=0,\) with \(0≤x≤4\). Find the flux across \(C\).
- Answer
- \(\text{flux}=-20\)
53. Let \(\vecs{F}=−y\,\mathbf{\hat i}+x\,\mathbf{\hat j}\) and let \(C: \vecs r(t)=\cos t\,\mathbf{\hat i}+\sin t\,\mathbf{\hat j}\) for \(0≤t≤2π\). Calculate the flux across \(C\).
54. Let \(\vecs{F}=(x^2+y^3)\,\mathbf{\hat i}+(2xy)\,\mathbf{\hat j}\). Calculate flux \(\vecs F\) orientated counterclockwise across the curve \(C: x^2+y^2=9.\)
- Answer
- \(\text{flux}=0\)
Complete the rest of the exercises as stated.
55. Find the line integral of \(\displaystyle\int _C z^2\,dx+y\,dy+2y\,dz,\) where \(C\) consists of two parts: \(C_1\) and \(C_2.\) \(C_1\) is the intersection of cylinder \(x^2+y^2=16\) and plane \(z=3\) from \((0, 4, 3)\) to \((−4,0,3).\) \(C_2\) is a line segment from \((−4,0,3)\) to \((0, 1, 5)\).
56. A spring is made of a thin wire twisted into the shape of a circular helix \(x=2\cos t,\;y=2\sin t,\;z=t.\) Find the mass of two turns of the spring if the wire has a constant mass density of \(ρ\) grams per cm.
- Answer
- \(m=4πρ\sqrt{5}\) grams
57. A thin wire is bent into the shape of a semicircle of radius \(a\). If the linear mass density at point \(P\) is directly proportional to its distance from the line through the endpoints, find the mass of the wire.
58. An object moves in force field \(\vecs F(x,y,z)=y^2\,\mathbf{\hat i}+2(x+1)y\,\mathbf{\hat j}\) counterclockwise from point \((2, 0)\) along elliptical path \(x^2+4y^2=4\) to \((−2,0)\), and back to point \((2, 0)\) along the \(x\)-axis. How much work is done by the force field on the object?
- Answer
- \(W=0\) units of work
59. Find the work done when an object moves in force field \(\vecs F(x,y,z)=2x\,\mathbf{\hat i}−(x+z)\,\mathbf{\hat j}+(y−x)\,\mathbf{\hat k}\) along the path given by \(\vecs r(t)=t^2\,\mathbf{\hat i}+(t^2−t)\,\mathbf{\hat j}+3\,\mathbf{\hat k}, \; 0≤t≤1.\)
60. If an inverse force field \(\vecs F\) is given by \(\vecs F(x,y,z)=\dfrac{k}{‖r‖^3}r\), where \(k\) is a constant, find the work done by \(\vecs F\) as its point of application moves along the \(x\)-axis from \(A(1,0,0)\) to \(B(2,0,0)\).
- Answer
- \(W=\frac{k}{2}\) units of work
61. David and Sandra plan to evaluate line integral \(\displaystyle\int _C\vecs F·d\vecs{r}\) along a path in the \(xy\)-plane from \((0, 0)\) to \((1, 1)\). The force field is \(\vecs{F}(x,y)=(x+2y)\,\mathbf{\hat i}+(−x+y^2)\,\mathbf{\hat j}\). David chooses the path that runs along the \(x\)-axis from \((0, 0)\) to \((1, 0)\) and then runs along the vertical line \(x=1\) from \((1, 0)\) to the final point \((1, 1)\). Sandra chooses the direct path along the diagonal line \(y=x\) from \((0, 0)\) to \((1, 1)\). Whose line integral is larger and by how much?
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .


1.28: Modification of the Analysis of Love
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Learning Objectives
- Explain the exception for when love does not require being good for the partner.
Watch this video or scan the QR code to learn ways to cope with your parents or in-laws becoming too involved in your relationship.
The train case points out an interesting flaw, however, in my analysis of love as it is stated. My criterion is that for a relationship to be one of reciprocal love, it must be that both people are good for each other. How good could it be for one to cause a train to run over his or her spouse? How good were the British officials being to their spouses when, if the story is true, they did not warn them to leave Coventry before the bombing? Yet we might, and I would, still want to say that the officials did love their families and would be loved by them even though they allowed harm to come to them, or allowed potential, yet avoidable, harm to befall them.
I wish to insert a necessary modification into my analysis of the ethical component or dimension of love, one which now can be understood without making the criteria seem difficult, unwieldy, or unnatural. The clause would exclude (from being unworthy of love) cases of causing or allowing harm (including even relatively minor things such as inconvenience or disappointment) to the loved or loving one if such harm is the result of overriding ethical obligations to others, or to self, when such overriding obligations are not unnecessarily brought on in the first place by the one doing the harm. For example, the Coventry case found those officials in that position; it was not as if they had known about it ahead of time and intentionally put their families in jeopardy. Consider the classic antagonistic mother-in-law, daughter-in-law conflict where each demands mutually exclusive behavior of the son/husband. A man, of course, does have obligations to his mother even after he is married, though he also has obligations to his wife as well. In the best of situations, those obligations will not be mutually exclusive too often, and both women will understand the cases when the husband feels obligated to opt for the other, understanding that he has treated them both as fairly as possible overall in such situations. Love will not be the worse for it. The worst situation is when both make mutually exclusive demands often and one or both do not understand ever being denied. In this case, probably neither will feel loving for very long even though it is not the man’s fault. But consider a slightly different case. Suppose a man continually places himself in such conflict by promising his mother things he does not have to promise, and which he would have no obligation to do if he did not make such promises. And suppose he continually hurts his wife’s feelings by keeping his promises to his mother and thus fulfilling his unnecessary and self-imposed obligations to his mother while thereby neglecting what would otherwise be his obligations to his wife. I think there is a point in saying that such a man is not as good to his wife as he could be and as he should be since he himself keeps unnecessarily bringing about the very conditions under which he is obligated to disappoint her or let her down. This would also be like the train case if the spouse at the switch had also been the one who first tied the other to the track.
One other potential problem with the ethical aspect of the analysis of love is that it is stated in terms of loving one’s good or welfare, not in terms of how it is right to treat them. I spent considerable time earlier showing that improving the welfare or good for the greatest number was not always ultimately the right thing to do, that there could be overriding factors. I am concerned that solely with regard to the loved one alone, and not because of some overriding obligation to another person or group, there might be some action which is right for them, but which diminishes their welfare or benefit, and not just temporarily (as when you have your ill child have an inoculation), but overall. I cannot think of any such cases but I cannot rule out the possibility. An example that comes close is that of punishing a child to teach it necessary behavior that all other means fail to teach. This is not a perfect example because teaching such behavior may be for the child’s own good in the long run anyway.
At any rate, if there are cases where diminishing another’s welfare is still for some reason the right way to treat them (and not just the right way for you to act because you have some overriding obligation to fulfill), then the ethics part of the analysis needs to be understood in terms of the lover’s being treated right rather than just having his or her good increased, bad decreased, or greatest amount of good over bad provided. If there are no cases where treating someone right is different from being good for them, then either “good” or “right” will do, but “right” is still the more general term.
Hence, the final version of the analysis would read: A loves B if and only if:
(1) A has strong feelings of attraction in general, or to some reasonable extent, for B,
(2) A, in general or to some reasonable extent, enjoys B (that is, A in general or to some reasonable extent is satisfied by B and by the things B does), particularly in areas of psychological importance (or meaningfulness) to A, and without particular disappointment or dissatisfaction in other such psychologically important (meaningful) areas, and
(3) the things B does are right for A, excluding cases where B diminishes A’s welfare because of acts B is obligated to do because of some overriding obligations (to self or others) which B did not unnecessarily or wrongfully bring on him or herself.
A and B would love each other if and only if, A loves B and B loves A; that is, if and only if they each had strong feelings of attraction for each other in general, enjoyed and satisfied each other in general, especially in areas of psychological importance (or meaningfulness), and in general did what was right for each other except in cases of overriding obligations to self or others that were not unnecessarily brought on by the one required to fulfill the obligation.
The briefer, originally stated version is more easily intelligible, more readily apparent, and in most cases is equally correct and useful.
Key Takeaways
- A modification on the author’s analysis of the ethical component or dimension of love is the clause that would exclude cases of causing or allowing harm to the loved or loving one if such harm is the result of overriding ethical obligations to others, or to self when such overriding obligations are not unnecessarily brought on in the first place by the one doing the harm.
- Another potential problem with the ethical aspect of the analysis of love is that it is stated in terms of loving one’s good or welfare, not in terms of how it is right to treat them.
Key Terms
- The classic antagonistic mother-in-law, daughter-in-law conflict where each demands mutually exclusive behavior of the son/husband.
Review Questions
- Question: What are the author’s criterion for: a relationship to be one of reciprocal love?
- Question : What is a potential problem with the initial ethical component of the analysis of love?


13.8: Downsizing and Morale
- Describe how to downsize without downsizing morale
As we discussed in the last section, downsizing often leads to frustration and insecurities. So if a company finds downsizing to be the most logical next step for their organization, how can the company minimize the damage layoffs can have on company morale? Unfortunately, there is no one-size-fits-all answer to this question. However, there are certain strategies and techniques companies can use to reduce backlash and salvage morale. The Human Resources Department is an essential part of a well-managed downsizing process. From start to finish, the HR team can help to properly communicate key messages, help former employees navigate the termination process, retrain and reallocate the remaining workforce, and fill many other important support roles. Check out the strategies below to learn more about ways to effectively handle downsizing and layoffs.
Communication is Key
Professional, informative, and clear communication is essential for ensuring a smooth downsizing process. Communication is extremely important in day-to-day interactions at work and it is especially important during confusing and trying times, like when a company is downsizing. Downsizing is a big challenge, for employers and employees alike. How a message is communicated and delivered can have either a positive or negative effect on the organization’s morale.
For example, if a company announces layoffs the day they begin terminations, they are going to get more backlash. Catching employees off guard with something that will greatly impact their livelihood will result in anger, fear, confusion, frustration, and other intense emotions and thoughts. Even employees who get to keep their jobs may feel caught off guard and lose any sense of security they have in their job. Too little communication can make it appear as though management is trying to hide something or that the plan is not well thought out and executed.
When there is clear communication around the downsizing plan and the reasons for downsizing, employees are more likely to be receptive to the message. There is no real way to sugarcoat a downsize or a termination; however, clear communication throughout the process is a respectful way for organizations to handle the situation.
A focus on communication should continue after layoffs occur. It is important to keep open lines of communication with the remaining employees and share the company’s vision for the future. It is also a good idea to address the elephant in the room, the layoffs. While it is important to move past the layoffs, it is equally as important to address them and not pretend as if they did not happen. Acknowledging change is necessary to move past the downsize and forward into the next chapter for the company. When management acts like nothing happened, it can create additional distrust and uneasiness among the remaining employees.
Provide Support
When companies layoff a large number of employees, it creates a gap between the workload and the workforce. Therefore, the remaining employees are needed to step up to fill the void and oftentimes are expected to take on greater responsibilities in addition to their current workload. This can be overwhelming for many employees. Providing support through resources, assistance, training, etc. can help to alleviate some of the frustrations and ensure that work is completed in a timely and high-quality manner.
Create a Fun and Engaged Work Environment
When employees see their coworkers getting laid off, it can spark insecurity and loneliness. People that employees were used to seeing on a daily basis will no longer be there, and some employees may be wondering if they will be the next to go. Hosting team building activities and events is a great way to promote teamwork and introduce employees to other coworkers they may not have worked with previously. When employees have a support team among their coworkers, they are more likely to be engaged and motivated. Creating fun events to foster mingling and team building, is a great way to help build healthy work relationships. A fun work environment can also help to show employee appreciation and demonstrate that the company is invested in their remaining employees.
It is important, however, that fun events are genuine, and not contrived. The motivation should be to build a strong team with the remaining employees, not simply a distraction from recent layoffs. Ensure there is a purpose behind fun activities and events (team building, networking, employee appreciation, etc.). Employees will be able to identify a work environment that is genuine versus one that is overcompensating to distract from recent events.
Respect and Understanding
Change impacts everyone differently. It is important to be understanding of the way individuals will handle the impacts of downsizing. Being respectful of the feelings of others, while still maintaining behavioral expectations, is important to rebuilding company morale. Managers should try to understand the root of employee frustrations and provide support and guidance when needed.
Once again, there is no “correct” way to handle downsizing and terminations. However, managers who use the strategies discussed in this section will be better equipped to maintain company morale during a difficult and trying time.
Contributors and Attributions
- Downsizing and Morale. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Authored by : josemiguels. Provided by : Pixabay. Located at : pixabay.com/illustrations/conversation-dialogue-interview-1262311/. License : CC0: No Rights Reserved . License Terms : Pixabay License
- Authored by : WOCinTech Chat. Provided by : Flickr. Located at : https://www.flickr.com/photos/wocintechchat/25900843762/ . License : CC BY: Attribution


10: Text
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As mentioned repeatedly, linguistic corpora, by their nature, consist of word forms, while other levels of linguistic representation are not represented unless the corresponding annotations are added. In written corpora, there is one level other than the lexical that is (or can be) directly represented: the text. Well-constructed linguistic corpora typically consist of (samples from) individual texts, whose meta-information (author, title, original place and context of publication, etc.) are known. There is a substantial body of corpus-linguistic research based on designs that combine the two inherently represented variables \(\mathrm{Word}\) (\(\mathrm{Form}\)) and \(\mathrm{Text}\); such designs may be concerned with the occurrence of words in individual texts, or, more typically, with the occurrence of words in clusters of texts belonging to the same language variety (defined by topic, genre, function, etc.).
Texts are, of course, produced by speakers, and depending on how much and what kind of information about these speakers is available, we can also cluster texts according to demographic variables such as dialect, socioeconomic status, gender, age, political or religious affiliation, etc. (as we have done in many of the examples in earlier chapters). In these cases, quantitative corpus linguistics is essentially a variant of sociolinguistics, differing mainly in that the linguistic phenomena it pays most attention to are not necessarily those most central to sociolinguistic research in general.


3.4: Half-planes
Assume \(X, Y \not\in (PQ)\). Then the angles \(PQX\) and \(PQY\) have the same sign if and only if \([XY]\) does not intersect \((PQ)\).
- Proof
-
The if-part follows from Lemma 3.3.2 .
Assume \([XY]\) intersects \((PQ)\); suppose that \(Z\) denotes the point of intersection. Without loss of generality, we can assume \(Z \ne P\).
Note that \(Z\) lies between \(X\) and \(Y\). According to Lemma 3.1.1 , \(\angle PZX\) and \(\angle PZY\) have opposite signs. It proves the statement if \(Z = Q\).
If \(Z \ne Q\), then \(\angle ZQX\) and \(QZX\) have opposite signs by 3.7. The same way we get that \(\angle ZQY\) and \(\angle QZY\) have opposite signs.
If \(Q\) lies between \(Z\) and \(P\), then by Lemma 3.1.1 two pairs of angles \(\angle PQX\), \(\angle ZQX\) and \(\angle PQY\), \(\angle ZQY\) have opposite signs. It follows that \(\angle PQX\) and \(\angle PQY\) have opposite signs as required.
In the remaining case \([QZ) = [QP)\) and therefore \(\angle PQX = \angle ZQX\) and \(\angle PQY = \angle ZQY\). Therefore again \(\angle PQX\) and \(\angle PQY\) have opposite signs as required.
The complement of a line \((PQ)\) in the plane can be presented in a unique way as a union of two disjoint subsets called half-planes such that
(a) Two points \(X, Y \not\in (PQ)\) lie in the same half-plane if and only if the angles \(PQX\) and \(PQY\) have the same sign.
(b) Two points \(X, Y \not\in (PQ)\) in the same half-plane if and only if \([XY]\) does not intersect \((PQ)\).
We say that \(X\) and \(Y\) lie on one side of \((PQ)\) if they lie in one of the half-planes of \((PQ)\) and we say that \(P\) and \(Q\) lie on the opposite sides of \(l\) if they lie in the different half-planes of \(l\).
Exercise \(\PageIndex{1}\)
Suppose that the angles \(AOB\) and \(A'OB'\) are vertical and \(B \not\in (OA)\). Show that the line \((AB)\) does not intersect the segment \([A'B']\).
- Hint
-
Note that \(O\) and \(A'\) lie on the same side of \((AB)\). Analogously \(O\) and \(B'\) lie on the same side of \((AB)\). Hence the result.
Consider the triangle \(ABC\). The segments \([AB], [BC]\), and \([CA]\) are called sides of the triangle.
Assume line \(l\) does not pass thru any vertex of a triangle. Then it intersects either two or zero sides of the triangle.
- Proof
-
Assume that the line \(l\) intersects side \([AB]\) of the triangle \(ABC\) and does not pass thru \(A, B,\) and \(C\).
By Corollary \(\PageIndex{1}\), the vertexes \(A\) and \(B\) lie on opposite sides of \(l\).
The vertex \(C\) may lie on the same side with \(A\) and on the opposite side with \(B\) or the other way around. By Corollary \(\PageIndex{1}\), in the first case, \(l\) intersects side \([BC]\) and does not intersect \([AC]\); in the second case, \(l\) intersects side \([AC]\) and does not intersect \([BC]\). Hence the statement follows.
Exercise \(\PageIndex{2}\)
Show that two points \(X, Y \not\in (PQ)\) lie on the same side of \((PQ)\) if and only if the angles \(PXQ\) and \(PYQ\) have the same sign.
- Hint
-
Apply Theorem 3.3.1 for \(\triangle PQX\) and \(\triangle PQY\) and then apply Corollary \(\PageIndex{1}\)a.
Exercise \(\PageIndex{3}\)
Let \(\triangle ABC\) be a nondegenerate triangle, \(A' \in [BC]\) and \(B' \in [AC]\). Show that the segments \([AA']\) and \([BB']\) intersect.
- Hint
-
We can assume that \(A' \ne B, C\) and \(B' \ne A, C\); otherwise the statement trivially holds.
Note that \((BB')\) does not intersect \([A'C]\). Applying Pasch's theorem (Theorem 3.4.1) for \(\triangle AA'C\) and \((BB')\), we get that \((BB')\) intersets \([AA']\); denote the point of intersection by \(M\).
The same way we get that \((AA')\) intersects \([BB']\); that is \(M\) lies on \([AA']\) and \([BB']\).
Exercise \(\PageIndex{4}\)
Assume that the points \(X\) and \(Y\) lie on opposite sides of the line \((PQ)\). Show that the half-line \([PX)\) does not intersect \([QY)\).
- Hint
-
Assume that \(Z\) is the point of intersection.
Note that \(Z \ne P\) and \(Z \ne Q\). Therefore, \(Z \in (PQ)\).
Show that \(Z\) and \(X\) lie on one side of \((PQ)\). Repeat the argument to show that \(Z\) and \(Y\) lie on one side of \((PQ)\). It follows that \(X\) and \(Y\) lie on the same side of \((PQ)\) - a contradiction.
Advanced Exercise \(\PageIndex{1}\)
Note that the follwing quantity
\[^{~} \measuredangle ABC = \begin{cases} \pi & \text{if } \measuredangle ABC = \pi \\ -\measuredangle ABC & \text{if } \measuredangle ABC < \pi \end{cases}\]
can serve as the angle measure; that is, the axioms hold if one exchanges \(\measuredangle\) to \(^{~} \measuredangle\) everywhere.
Show that \(\measuredangle\) and \(^{~} \measuredangle\) are the only possible angle measures on the plane.
Show that without
Axiom IIIc
, this is no longer true.


9.6: Keep or Replace a Fixed Asset
A company may consider replacing a fixed asset it currently owns and operates with one that is more efficient to reduce operating costs. The company can use differential analysis to compare the cost of keeping its original asset and the cost of replacing it with something new.
McNamara Company is considering selling an existing piece of equipment for $75,000 and replacing it with new equipment that will cost $182,000. The old equipment cost $200,000 and currently has accumulated depreciation of $120,000. The estimated annual variable operational costs for the next six years are $21,000 for the old equipment and $6,000 for the new equipment.
|
Keep |
Replace |
Difference |
|
|
Revenue |
$75,000 |
||
|
Costs: |
|||
|
Purchase price |
182,000 |
||
|
Operating costs 1 |
$126,000 |
36,000 |
|
|
Income (loss) |
($126,000) |
($143,000) |
$17,000 |
1 $21,000 x 6 \(\ \quad \quad\) $6,000 x 6
Replacing the equipment will generate revenue from the sale of the original asset, but the cost of acquiring the new asset will also be incurred. Each alternative has different annual operating costs. The purchase price of the original equipment is a sunk cost that will not be recouped or changed regardless of the decision; therefore, it is ignored. Each alternative independently generates a loss, so the choice with the lower loss is preferable. The company should keep and operate the original equipment.


11.3: Oxygenation Equipment
There are several types of equipment a nurse may use when providing oxygen therapy to a patient. Each device is described in detail below.
Pulse Oximeter
A pulse oximeter is a commonly used portable device used to obtain a patient’s oxygen saturation level at the bedside or in a clinic. See Figure \(\PageIndex{1}\) [1] for an image of a portable pulse oximeter. The pulse oximeter, commonly referred to as a “Pulse Ox,” is an electronic device that measures the oxygen saturation of hemoglobin in a patient’s red blood cells, referred to as SpO2. The normal range for SpO2 for an adult without an underlying respiratory condition is above 92%. The pulse oximeter analyzes light produced by the probe as it passes through the finger to determine the saturation level of the hemoglobin molecule. Pulse oximeter probes can be attached to a patient’s finger, forehead, nose, foot, ear, or toe. However, pulse oximetry readings can be inaccurate if the patient is wearing nail polish, has reduced perfusion of the extremities due to a cardiovascular condition, or has other molecules attached to hemoglobin such as in the case of carbon monoxide poisoning. [2]
Oxygen Flow Meter
In inpatient settings, rooms are equipped with wall-mounted oxygen supply outlets that are nationally standardized in a green color, whereas air outlets are standardized with a yellow color. Oxygen flow meters are attached to the green oxygen outlets, and then the oxygenation device is attached to the flow meter. See Figure \(\PageIndex{2}\) [3] for an image of an oxygen flow meter. An oxygen flow meter consists of a glass cylinder containing a steel ball with an opening through which oxygen from the supply source is injected through an adapter. This adapter is commonly referred to as a “tree” because of its appearance. Oxygen is turned on, and the flow rate of oxygen is controlled by turning the green valve on the side of the glass cylinder. The flow rate is set according to the location of a steel ball inside the cylinder and the numbered lines on the glass cylinder. For example, in Figure \(\PageIndex{2}\), the flow rate is currently set at 2 liter per minute (L/min). It is essential to implement safety precautions whenever oxygen is used. Read more about “Safety with Oxygen Therapy” later in this section.
Portable Oxygen Supply Devices
Portable oxygen tanks are commonly used when transporting a patient to procedures within the hospital or to other agencies. See Figure \(\PageIndex{3}\) [4] for an image of a portable oxygen tank. Oxygenation devices are connected to the tank in a similar manner as the wall-mounted oxygen flow meter. It is crucial for nurses and transporters to ensure the tank has an adequate amount of oxygen for use during transport, is turned on, and the appropriate flow rate is set.
Instead of oxygen tanks, oxygen concentrators are commonly used by patients in their home environment. See Figure \(\PageIndex{4}\) [5] for an image of a home oxygen concentrator. Oxygen concentrators are also produced in portable sizes that are lightweight and easy for patient use while travelling and mobile in the community. See Figure \(\PageIndex{5}\) [6] for an image of a portable oxygen concentrator. Oxygen concentrators work by taking the 21% concentration of oxygen in the air, running it through a molecular sleeve to remove the nitrogen and concentrating the oxygen to a 96% level, thus producing between 1 and 6 liters per minute of oxygen. Oxygen concentrators may provide pulse flow or continuous flow. Pulse flow only occurs on inhalation, whereas continuous flow delivers oxygen throughout the entire breath cycle. Pulse versions are the most lightweight because oxygen is provided only as needed by the patient. [7]
Nasal Cannula
A nasal cannula is the simplest oxygenation device and consists of oxygen tubing connected to two short prongs that are inserted into the patient’s nares. See Figure \(\PageIndex{6}\) [8] for an image of a nasal cannula. The tubing is connected to the flow meter of the oxygen supply source. To prevent drying out the patient’s mucus membranes, humidification may be added for hospitalized patients receiving oxygen flow rates greater than 4 L/minute or for those receiving oxygen therapy for longer periods of time. [9]
Nasal cannulas are the most common type of oxygen equipment. They are used for short- and long-term therapy (i.e., COPD patients) and are best used with stable patients who require low amounts of oxygen.
Flow rate: Nasal cannulas can have a flow rate ranging from 1 to 5 liters per minute (L/min), with a 4% increase in FiO2 for every liter of oxygen, resulting in range of fraction of inspired oxygen (FiO2) levels of 24-44%.
Advantages: Nasal cannulas are easy to use, inexpensive, and disposable. They are convenient because the patient can talk and eat while receiving oxygen.
Limitations: The nasal prongs of nasal cannula are easily dislodged, especially when the patient is sleeping. The tubing placed on the face can cause skin breakdown in the nose and above the ears, so the nurse must vigilantly monitor these areas. Based on agency policy, the nurse should add padding to the oxygen tubing as needed to avoid skin breakdown and may apply a water-based lubricant to prevent drying. However, petroleum-based lubricant should not be used due to the risk of flammability. Nasal cannulas are not as effective if the patient is a mouth breather or has blocked nostrils, a deviated septum, or nasal polyps. [10]
High-Flow Nasal Cannula
High-flow nasal cannula therapy is an oxygen supply system capable of delivering up to 100% humidified and heated oxygen at a flow rate of up to 60 liters per minute. [11] Patients with high-flow nasal cannulas are generally in critical condition and require advanced monitoring. See Figure \(\PageIndex{7}\) [12] for an illustration of a high-flow nasal cannula system that is initially set up by a respiratory therapist and then maintained by a nurse.
Simple Mask
A simple mask fits over the mouth and nose of the patient and contains exhalation ports (i.e., holes on the side of the mask) through which the patient exhales carbon dioxide. These holes should always remain open. The mask is held in place by an elastic band placed around the back of the head. It also has a metal piece near the top that can be pinched and shaped over the patient’s nose to create a better fit. Humidified air may be attached if the oxygen concentrations are drying for the patient. See Figure \(\PageIndex{8}\) [13] for an image of a simple face mask.
Flow Rate: Simple masks should be set to a flow rate of 6 to 10 L/min, resulting in oxygen concentration (FiO2) levels of 35%-50%. The flow rate should never be set below 6 L/min because this can result in the patient rebreathing their exhaled carbon dioxide.
Advantages: Face masks are used to provide moderate oxygen concentrations. Their efficiency in oxygen delivery depends on how well the mask fits and the patient’s respiratory demands.
Disadvantages: Face masks must be removed when eating, and they may feel confining for some patients who feel claustrophobic with the mask on. [14]
Non-Rebreather Mask
A non-rebreather mask consists of a mask attached to a reservoir bag that is attached with tubing to a flow meter. See Figure \(\PageIndex{9}\) [15] for an image of a non-rebreather mask. It has a series of one-way valves between the mask and the bag and also on the covers on the exhalation ports. The reservoir bag should never totally deflate; if the bag deflates, there is a problem and immediate intervention is required. The one-way valves function so that on inspiration, the patient only breathes in from the reservoir bag; on exhalation, carbon dioxide is directed out through the exhalation ports. Non-rebreather masks are used for patients who can breathe on their own but require higher concentrations of oxygen to maintain satisfactory blood oxygenation levels.
Flow rate: The flow rate for a non-rebreather mask should be set to deliver a minimum of 10 to 15 L/minute. The reservoir bag should be inflated prior to placing the mask on the patient. With a good fit, the non-rebreather mask can deliver between 60% and 80% FiO2.
Advantages: Non-rebreather masks deliver high levels of oxygen noninvasively to patients who can otherwise breathe unassisted.
Disadvantages: Due to the one-way valves in non-rebreather masks, there is a high risk of suffocation if the gas flow is interrupted. The mask requires a tight seal and may feel hot and confining to the patient. It will interfere with talking, and the patient cannot eat with the mask on.
Partial Rebreather Mask
The partial rebreather mask looks very similar to the non-rebreather mask. The difference between the masks is a partial rebreather mask does not contain one-way valves, so the patient’s exhaled air mixes with their inhaled air. A partial rebreather mask requires 10-15 L/min of oxygen, but only delivers 35-50% FiO2.
Venturi Mask
Venturi masks are indicated for patients who require a specific amount of supplemental oxygen to avoid complications, such as those with chronic obstructive pulmonary disease (COPD). Different types of adaptors are attached to a face mask that set the flow rate to achieve a specific FiO2 ranging from 24% to 60%. Venturi adapters are typically set up by a respiratory therapist, but in some facilities they may be set up by a nurse according to agency policy.
Flow rate: The flow rate depends on the adaptor and does not correspond to the flow meter. Consult with a respiratory therapist before changing the flow rate.
Advantages: A specific amount of FiO2 is delivered to patients whose breathing status may be affected by high levels of oxygen.
Continuous Positive Airway Pressure (CPAP)
A continuous positive airway pressure (CPAP) device is used for people who are able to breathe spontaneously on their own but need help in keeping their airway unobstructed, such as those with obstructive sleep apnea. (See Table 11.2.3 in the “Basic Concepts of Oxygenation” section for more information about obstructive sleep apnea.) The CPAP device consists of a special mask that covers the patient’s nose, or nose and mouth, and is attached to a machine that continuously applies mild air pressure to keep the patient’s airways from collapsing.
A prescription is required for a CPAP device in the hospital or patient’s home environment. In the hospital, the FiO2 is set up with the CPAP mask by the respiratory therapist. In a home setting, an adapter is added so that oxygen is attached using a flowmeter with preprogrammed settings so the patient and/or nurse are only required to turn the machine on before sleeping and off upon awakening. It is important to keep the mask and tubing clean to prevent infection, so be sure to follow agency policy for cleaning the equipment regularly. If a humidifier is attached, distilled water or sterile water should be used to fill it, but never tap water. See Figure \(\PageIndex{10}\) [16] for an illustration of a patient wearing a CPAP device while sleeping.
BiPAP
A Bilevel Positive Airway Pressure (BiPAP) device is similar to a CPAP device in that it is used to prevent airways from collapsing, but BiPAP devices have two pressure settings. One setting occurs during inhalation and a lower pressure setting is used during exhalation. Patients using BiPAP devices in their home environment for obstructive sleep apnea often find these two pressures more tolerable because they don’t have to exhale against continuous pressure. In acute-care settings, BiPAP devices are also used for patients in acute respiratory distress as a noninvasive alternative to intubation and mechanical ventilation and are managed by respiratory therapists. BiPAP devices in home settings are set up in a similar manner as CPAP machines for ease of use. See Figure \(\PageIndex{11}\) [18] for an image of a simulated patient wearing a BiPAP mask in a hospital setting with continuous pulse oximetry monitoring.
Bag Valve Mask (Ambu Bag)
A bag valve mask, commonly known as an “Ambu bag,” is a handheld device used in emergency situations for patients who are not breathing (respiratory arrest) or who are not breathing adequately (respiratory failure). In this manner, this device is different from the other devices because it assists with ventilation , the movement of air into and out of the lungs, as well as oxygenation. See Figure \(\PageIndex{12}\) [19] for an image of a bag valve mask. Bag valve masks are produced in different sizes for infants, children, and adults to prevent lung injury, so it is important to use the correct size for the patient.
When using a bag mask valve, the rescuer manually compresses the bag to force air into the lungs. Squeezing the bag once every 5 to 6 seconds for an adult or once every 3 seconds for an infant or child provides an adequate respiratory rate. In inpatient settings, the bag mask valve is attached to an oxygen supply to increase the concentration of oxygenation provided with each breath. See Figure \(\PageIndex{13}\) [20] for an illustration of operating a bag valve mask.
It is vital to obtain a tight seal of the mask to the patient’s face, but this is difficult for a single rescuer to achieve. Therefore, two rescuers are recommended; one rescuer performs a jaw thrust maneuver, secures the mask to the patient’s face with both hands, and focuses on maintaining a leak-proof mask seal, while the other rescuer squeezes the bag and focuses on the amount and the timing.
Flow rate: The flow rate for a bag valve mask attached to an oxygen source should be set to 15 L/minute, resulting in FiO2 of 100%.
Advantages: A bag valve mask is portable and provides immediate assistance to patients in respiratory failure or respiratory arrest. It also can be used to hyperoxygenate patients before procedures that can cause hypoxia, such as tracheal suctioning.
Disadvantages: The rate and depth of compression of the bag must be closely monitored to prevent injury to the patient. In the event of respiratory failure when the patient is still breathing, the bag compressions must be coordinated with the patient’s inhalations to ensure that oxygen is delivered and asynchrony of breaths is prevented. Complications may also result from overinflating or overpressurizing the patient. Complications include lung injury or the inflation of the stomach that can lead to aspiration of stomach contents. Additionally, rescuers may tire after a few minutes of manually compressing the bag, resulting in less than optimal ventilation. Alternatively, an endotracheal tube (ET) can be inserted by an advanced practitioner to substitute for the mask portion of this device. See more information about endotracheal tubes below.
Endotracheal Intubation
When a patient is receiving general anesthesia prior to a procedure or surgery or is experiencing respiratory failure or respiratory arrest, an endotracheal tube (ET) is inserted by an advanced practitioner, such as a respiratory therapist, paramedic, or anesthesiologist, to maintain a secure airway. The ET tube is sealed within the trachea with an inflatable cuff, and oxygen is supplied via a bag valve mask or via mechanical ventilation. See Figure \(\PageIndex{14}\) [21] for an image of a cuffed endotracheal tube.
Mechanical Ventilator
A mechanical ventilator is a machine attached to an endotracheal tube to assist or replace spontaneous breathing. Mechanical ventilation is termed invasive because it requires placement of a device inside the trachea through the mouth, such as an endotracheal tube. Mechanical ventilators are managed by respiratory therapists via protocol or provider order. FiO2 can be set from 21-100%. Nurses collaborate with respiratory therapists and the health care providers regarding the overall care of the patient on a mechanical ventilator. See Figure \(\PageIndex{15}\) [22] for an image of a simulated patient who is intubated with an endotracheal tube and attached to a mechanical ventilator.
 A tracheostomy tube is placed through the stoma and directly into the trachea to maintain an open (patent) airway and to administer oxygen. A tracheostomy may be performed emergently or as a planned procedure. Read more about tracheostomies in the “ Tracheostomy Care and Suctioning ” chapter.
Flow Rates and Oxygen Percentages
When administering oxygen to a patient, it is important to ensure that oxygen flow rates are appropriately set according to the type of administration device. Review Table \(\PageIndex{1}\) to review appropriate settings for various types of oxygenation devices.
| Device | Flow Rates and Oxygen Percentage |
|---|---|
| Nasal Cannula | Flow rate: 1-6 L/min |
| Bag Valve Mask |
Flow rate: 15 L/min
FiO2: 100% Squeeze the bag once every 5 to 6 seconds for an adult or once every 3 seconds for an infant or child. |
Safety with Oxygen Therapy
Oxygen therapy supports life, but it also supports fire. While there are many benefits to oxygen therapy, there are also many hazards. Oxygen must be administered cautiously and according to the safety guidelines in Table \(\PageIndex{2}\). [23]
| Guideline | Additional Information |
|---|---|
| Remember that oxygen is a medication. | Oxygen is a medication and should not be adjusted without consultation with a physician or respiratory therapist. |
| Store oxygen cylinders correctly. | When using oxygen cylinders, store them upright, chained, or in appropriate holders so that they will not fall. Full oxygen tanks should be stored separately from partially-full or empty oxygen tanks. |
| Use tank holders appropriately. | When transporting a patient, proper tank holders must be used per Joint Commission guidelines. Tanks should never be placed on the patient’s bed. |
| Do not allow smoking near the oxygen devices. | Oxygen supports combustion. No smoking is permitted around any oxygen delivery devices in the hospital or home environment. |
| Keep oxygen cylinders away from heat sources. | Keep oxygen delivery systems at least 5 feet from any heat source. |
| Check for electrical hazards in the home or hospital prior to use. | Determine that electrical equipment in the room or home is in safe working condition. A small electrical spark in the presence of oxygen will result in a serious fire. The use of a gas stove, kerosene space heater, or smoker is unsafe in the presence of oxygen. Avoid items that may create a spark (e.g., electrical razor, hair dryer, synthetic fabrics that cause static electricity, or mechanical toys) with nasal cannula in use. Petroleum-based lubricants should not be used on the lips or around the nasal cannula. |
| Check levels of oxygen in portable tanks. | Check oxygen levels of portable tanks before transporting a patient to ensure that there is enough oxygen in the tank. |
- “ OxyWatch_C20_Pulse_Oximeter.png ” by Thinkpaul is licensed under CC BY-SA 3.0 ↵
- American Lung Association. (2020, May 27). Pulse oximetry . https://www.lung.org/lung-health-diseases/lung-procedures-and-tests/pulse-oximetry ↵
- “Oxygen Regulator3I3A1063.jpg” by Deanna Hoyord, Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- “ RECALLED_–_Portable_oxygen_cylinder_units_(8294127015).jpg ” by The U.S. Food and Drug Administration is in the Public Domain ↵
- “ Invacare Perfecto 2 Oxygen Concentrator.JPG ” by BrokenSphere / Wikimedia Commons is licensed under CC BY-SA 3.0 ↵
- “ Portable Oxygen Concentrator by Inogen.jpg” by Oxystore is licensed under CC BY-SA 4.0 ↵
- Gibson, C. M. (Ed.). Portable oxygen concentrator. WikiDoc. https://www.wikidoc.org/index.php/Portable_oxygen_concentrator ↵
- “Image00011.jpg” by British Columbia Institute of Technology is licensed under CC BY 4.0. Access for free at https://opentextbc.ca/clinicalskills/chapter/5-5-oxygen-therapy-systems/ ↵
- Duck, A. (2009, December 14). Does oxygen need humidification? https://www.nursingtimes.net/clinical-archive/respiratory-clinical-archive/does-oxygen-need-humidification-14-12-2009/ ↵
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology and is licensed under CC BY 4.0 ↵
- This work is a derivative of StatPearls by Sharma, Danckers, Sanghavi, and Chakraborty and is licensed under CC BY 4.0 ↵
- “ HFT diagram.png ” by Strangecow is in the Public Domain ↵
- “DSC_2086.jpg” by British Columbia Institute of Technology is licensed under CC BY 4.0. Access for free at https://opentextbc.ca/clinicalskills/chapter/5-5-oxygen-therapy-systems/ ↵
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology is licensed under CC BY 4.0 ↵
- “DSC_2083.jpg” by British Columbia Institute of Technology is licensed under CC BY 4.0. Access for free at https://opentextbc.ca/clinicalskills/chapter/5-5-oxygen-therapy-systems/ ↵
- “ Depiction of a sleep apnea patient using a CPAP machine ” by https://www.myupchar.com/en is licensed under CC BY 4.0 ↵
- U.S. Food and Drug Administration. (2012, December 12). CPAP tips from FDA. [Video]. YouTube. https://youtu.be/B10ABypyGOo ↵
- “Simulated patient wearing a BiPAP mask” by Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- “ Ballon ventilation 1.jpg ” by Rama is licensed under CC BY-SA 2.0 FR ↵
- “ Resuscitator 3 - Operation (PSF).png ” by Pearson Scott Foresman is in the Public Domain ↵
- “ Sondeintubation.jpg ” by bigomar2 is licensed under CC BY-SA 3.0 ↵
- “Simulated intubated patient on a mechanical ventilator” by ARISE project is licensed under CC BY 4.0 ↵
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology and is licensed under CC BY 4.0 ↵


1.19: Second Declension Nouns
Module 16
© 2021 Philip S. Peek, CC BY 4.0 https://doi.org/ 10.11647/OBP.0264. 19
Nouns
Nouns in Greek are defined just like nouns are in English; but the way they create meaning is different. As in English, Greek nouns ( ὀνόματα ) refer to people, places, things, and ideas. Greek nouns have endings. English nouns can change form when they show possession, as in Jada’s book , where the ’s is added as a suffix and indicates that the book belongs to Jada. English nouns also change form when expressing the plural: two suns , three oxen , four mice . The endings on Greek nouns, as we have seen previously, create the same meanings that English does through form change, word order, and the use of prepositional phrases.
Second Declension Nouns
The nominative singular of nouns of the second declension ends in - ος or – ον . Most second declension nouns whose nominative ends in - ος are masculine in gender and a few are feminine. Second declension nouns whose nominative ending is – ον are neuter in gender. Memorize these endings, know how to obtain the noun’s stem, and know how to decline the nouns.
Declining Second Declension Nouns in -ος or -ον
To decline second declension noun endings in - ος or – ον , first get the stem by removing the genitive singular ending. For sets 7 and 8, the genitive singular ending is - ου . For sets 9 and 10, the genitive singular ending is - ος . For sets 1 and 2, the genitive singular is - ης or - ᾱς . What remains is the stem. To the stem add the following endings:
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|
||||||||||||||||||||||||||||||||||||||||||||||||
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|
ἥλιος, ἡλίου, ἀδελφός, ἀδελφοῦ, ἔργον, ἔργου
To decline the noun ἥλιος , ἡλί ου ὁ sun , take the genitive singular ἡλίου , remove the genitive singular ending - ου to get the stem: ἡλί -, and add the masculine/feminine endings from above.
To decline ἀδελφό ς , ἀδελφοῦ ὁ brother , take the genitive singular ἀδελφοῦ , remove the genitive singular ending - ου to get the stem: ἀδελφ - , and add the masculine/feminine endings from above.
To decline ἔργον , ἔργου τ ό work , take the genitive singular ἔργου , remove the genitive singular ending - ου to get the stem: ἔργ - , and add the neuter endings from above.
|
Singular |
|||
|
N |
ἥλιος |
ἀδελφός |
ἔργον |
|
A |
ἥλιον |
ἀδελφόν |
ἔργον |
|
G |
ἡλίου |
ἀδελφοῦ |
ἔργου |
|
D |
ἡλίῳ |
ἀδελφῷ |
ἔργῳ |
|
V |
ἥλιε |
ἄδελφε |
ἔργον |
|
Plural |
|||
|
N |
ἥλιοι |
ἀδελφοί |
ἔργα |
|
A |
ἡλίους |
ἀδελφούς |
ἔργα |
|
G |
ἡλίων |
ἀδελφῶν |
ἔργων |
|
D |
ἡλίοις |
ἀδελφοῖς |
ἔργοις |
|
V |
ἥλιοι |
ἀδελφοί |
ἔργα |
|
Singular |
|||
|
N |
ἥλιος |
ἀδελφός |
ἔργον |
|
G |
ἡλίου |
ἀδελφοῦ |
ἔργου |
|
D |
ἡλίῳ |
ἀδελφῷ |
ἔργῳ |
|
A |
ἥλιον |
ἀδελφόν |
ἔργον |
|
V |
ἥλιε |
ἄδελφε |
ἔργον |
|
Plural |
|||
|
N |
ἥλιοι |
ἀδελφοί |
ἔργα |
|
G |
ἡλίων |
ἀδελφῶν |
ἔργων |
|
D |
ἡλίοις |
ἀδελφοῖς |
ἔργοις |
|
A |
ἡλίους |
ἀδελφούς |
ἔργα |
|
V |
ἥλιοι |
ἀδελφοί |
ἔργα |
ἥλιος:
- Note that the vocative singular is different from the nominative singular. As in all nouns, the vocative plural is the same as the nominative plural.
- The accent does not shift to the ultima in the genitive plural as it does in the first declension.
- Final - αι and - οι count as short for purposes of accentuation except in the optative, a mood that you’ll learn in Part II of the 21st-Century series. Thus the accent remains on the antepenult in ἥλιοι .
ἀδελφός:
- When first and second declension nouns have an acute accent on the ultima in the nominative singular, the accent changes to a circumflex in the genitive and dative, singular and plural.
- ἄδελφε is an exception for the rules of accent.
ἔργον:
- In all neuter nouns, the accusative and vocative are the same as the nominative, both in the singular and in the plural.
- The nominative and vocative plural ending of all neuter nouns is - α .
Practice Translating. Translate the sentences below adapted from the prologue of Euripides’ Bakkhai ( Βάκχαι ). Remember the meanings and functions of the cases presented in Module 7 . Nominative case endings are bolded; genitive endings are italicized; dative endings are highlighted; and accusative endings are underlined. Note that the third declension increases the number of possible endings for the nominative singular. Check your understanding with the translations in the Answer Key . Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
|
Case |
Ending |
Function |
|
Nominative |
- α , - η , - ης , ις , - ξ , - ο , - ον , - ος , - ρ , - ς , - ων |
subject of the verb |
|
Genitive |
- ας , - ης , - ος , - ου , - ων |
dependence; object of preposition; possession |
|
Dative |
- οις , - σι |
dative with adjective |
|
Accusative |
- α , - αν , - ας , - ε , - εις , - η , - ην , - ν , - ο , - ον , - ος |
motion toward; object of verb or preposition |
ἥκω Δι ὸς παῖ ς τ ὴν Θηβαί ων χθόν α , Διόνυσ ος . ἐμ ὲ μὲν τίκτει ποθ ’ ἡ Κάδμ ου κόρ η , Σεμέλ η · ἐμ ὲ δὲ ἀστραπηφόρ ον λοχεύει πῦ ρ . μορφ ὴν δʼ ἀμείβω ἐκ θε οῦ βροτεί αν · ὁρ άω δὲ μητρ ὸς μνῆμ α καὶ δό μ ων ἐρείπι α . τ ὰ δὲ ἔτι τύ φεται , Δί ου πυρ ὸς φλό ξ , ἀθάνατ ος Ἥρ ας μητέρ α εἰς ἐμ ὴν ὕβρ ις . αἰνέω δὲ Κάδμ ον . ὁ δὲ γὰρ ἄβατ ον τ ὸ πέδ ον τίθησι , θυγατρ ὸς σηκ όν . λείπω δὲ ἐγὼ Λυδ ῶν γύ ας Φρυγ ῶν τε , Περσ ῶν τε πλάκ ας Βάκτρι ά τε τείχ η τ ήν τε χθόν α Μήδ ων . καὶ δὲ ἐ πέρχομαι Ἀραβί αν τ ’ Ἀσί αν τε . ἡ δὲ Ἀσί α παρ ’ ἅλ α κεῖται καὶ ἔχει μιγάδ ας Ἕλλη σι βαρβάρ οις θʼ ὁμοῦ πόλ εις . ἐκεῖ μὲν ἤδη χορεύω καὶ καθίστημι τ ὰς τελετ άς . ἐγὼ δὲ ν ῦν ἔρχομαι εἰς Ἑλλήν ων πόλι ν ὅτι εἰμὶ ἐμφαν ὴς δαίμ ων βροτ οῖς . πρώτ ας δὲ θήβ ας τ ῆς γ ῆς Ἑλληνίδ ος ἀνολολύζω , καὶ νεβρ ίδ α ἐξάπτω χρο ὸς θύρσ ον τε δίδωμι ἐς χεῖρ α , κίσσιν ον βέλ ος .
Adverbs and Verbs
|
αἰνέω praise, approve, promise |
* καθίστημι appoint, establish, put into a state ; (intrans.) be established, be appointed |
|
ἀμείβω answer; change, exchange |
* κεῖμαι lie |
|
ἀνολολύζω cry aloud, shout; excite |
λείπω leave |
|
* δίδωμι give ; δίκην δίδωμι I pay the penalty ; δίδωμι χάριν I give thanks |
λοχεύω bring forth, bear; deliver |
|
* εἰμί be |
* ὁράω see |
|
ἐ κεῖ there |
* τίθημι put, place |
|
ἐξάπτω fasten to or from |
τίκτω bear, give birth |
|
ἐπέρχομαι come upon; approach; attack |
τύφω raise a smoke, smoke ; (pass.) smolder |
|
* ἔρχομαι come, go |
χορεύω dance |
|
* ἥκω have come, be present |
Adjectives, Nouns, Pronouns
|
Νοminative |
Genitive |
Dative |
Accusative |
English Equivalent |
|
ἄβατον |
ἀβάτου |
ἀβάτῳ |
ἄβατον |
untrodden, impassable |
|
ἀθάνατος |
ἀθανάτου |
ἀθανάτῳ |
ἀθάνατον |
immortal, deathless |
|
ἅλς |
ἁλός |
ἁλί |
ἅλα |
sea |
|
Ἀραβία |
Ἀραβίας |
Ἀραβίᾳ |
Ἀραβίαν |
Arabia |
|
Ἀσία |
Ἀσίας |
Ἀσίᾳ |
Ἀσίαν |
Asia |
|
ἀστραπηφόρον |
- φόρου |
- φόρῳ |
- φόρον |
lightning-bearing |
|
Βάκτρια |
Βακτρίων |
Βακτρίοις |
Βάκτρια |
Baktria |
|
βάρβαροι |
βαρβάρων |
βαρβάροις |
βαρβάρους |
barbarians |
|
βέλος |
βέλεος (- ους ) |
βέλει |
βέλος |
missile, arrow, dart |
|
βροτεία |
βροτείας |
βροτείᾳ |
βροτείαν |
mortal |
|
βροτοί |
βροτῶν |
βροτοῖς |
βροτούς |
mortals |
|
* γῆ |
γῆς |
γῇ |
γῆν |
land, earth |
|
γύαι |
γυῶν |
γύαις |
γύας |
lands |
|
δαίμων |
δαίμονος |
δαίμονι |
δαίμονα |
god, deity |
|
Διόνυσος |
Διονύσου |
Διονύσῳ |
Διόνυσον |
Dionysos |
|
δόμοι |
δόμων |
δόμοις |
δόμους |
house, houses |
|
* ἐ γώ |
ἐμοῦ |
ἐμοί |
ἐμέ |
I, me, mine |
|
* Ἕλληνες |
Ἑλλήνων |
Ἕλλησι ( ν ) |
Ἕλληνας |
Greeks |
|
Ἑλληνίς |
Ἑλληνίδος |
Ἑλληνίδι |
Ἑλληνίδα |
Greek |
|
* ἐμ ή |
ἐμῆς |
ἐμῇ |
ἐμήν |
my |
|
ἐμφανή ς |
ἐμφανέος (- οῦς ) |
ἐμφανεῖ |
ἐμφανέα (- ῆ ) |
clear, manifest |
|
ἐρείπιον |
ἐρειπίου |
ἐρειπίῳ |
ἐρείπιον |
ruin, wreck |
|
* Ζεύς |
Διός ( Ζηνός ) |
Διί ( Ζηνί ) |
Δία ( Ζῆνα ) |
Zeus |
|
Ἥρα |
Ἥρας |
Ἥρᾳ |
Ἥραν |
Hera |
|
* θεός |
θεοῦ |
θεῷ |
θεόν |
god, goddess |
|
θῆβαι |
θηβῶν |
θήβαις |
θήβας |
Thebes |
|
Θηβαῖοι |
Θηβαίων |
Θηβαίοις |
Θηβαίους |
Theban |
|
θυγάτηρ |
θυγατρός |
θυγατρί |
θυγατέρα |
daughter |
|
θύρσος |
θύρσου |
θύρσῳ |
θύρσον |
thyrsos |
|
Κάδμος |
Κάδμου |
Κάδμῳ |
Κάδμον |
Kadmos |
|
κίσσινος |
κισσίνου |
κισσίνῳ |
κίσσινον |
of ivy |
|
κόρη |
κόρης |
κόρῃ |
κόρην |
girl; daughter |
|
Λυδοί |
Λυδῶν |
Λυδοῖς |
Λυδούς |
Lydian |
|
Μῆδοι |
Μήδων |
Μήδοις |
Μήδους |
Mede, Persian |
|
* μήτηρ |
μητρός |
μητρί |
μητέρα |
mother |
|
μιγάδες |
μιγάδων |
μιγάσι ( ν ) |
μιγάδας |
mixed |
|
μνῆμα |
μνήματος |
μνήματι |
μνῆμα |
memorial, record, tomb |
|
μορφή |
μορφῆς |
μορφῇ |
μορφήν |
form, appearance |
|
νεβρίς |
νεβρίδος |
νεβρίδι |
νεβρίδα |
fawnskin |
|
* παῖς |
παιδός |
παιδί |
παῖδα |
child |
|
πέδον |
πέδου |
πέδῳ |
πέδον |
earth, ground |
|
Πέρσαι |
Περσέων (- ῶν ) |
Πέρσαις |
Πέρσας |
Persians |
|
πλάκες |
πλακῶν |
πλαξί ( ν ) |
πλάκας |
plains, fields |
|
* πόλις |
πόλεως |
πόλει |
πόλιν |
city |
|
πρ ῶται |
πρώτων |
πρώταις |
πρώτας |
first |
|
πῦρ |
πυρός |
πυρί |
πῦρ |
fire |
|
Σεμέλη |
Σεμέλης |
Σεμέλῃ |
Σεμέλην |
Semele |
|
σηκός |
σηκοῦ |
σηκῷ |
σηκόν |
pen, fold, precinct |
|
τά δέ |
τῶν δέ |
τοῖς δέ |
τά δέ |
they, them, theirs |
|
τείχη |
τειχῶν |
τείχεσι ( ν ) |
τείχη |
walls |
|
τελεταί |
τελετῶν |
τελεταῖς |
τελετάς |
rites, mysteries |
|
ὕβρις |
ὕβρεως |
ὕβρει |
ὕβριν |
hybris, brutality |
|
φλόξ |
φλογός |
φλογί |
φλόγα |
flame, fire, blaze |
|
Φρύγες |
Φρυγῶν |
Φρυξί ( ν ) |
Φρύγας |
Phrygian |
|
* χείρ |
χειρός |
χειρί |
χε ῖρα |
hand; force, army |
|
χθών |
χθονός |
χθονί |
χθόνα |
earth, ground |
|
χρώς |
χρωτός ( χροός ) |
χρωτί ( χροΐ ) |
χρῶτα ( χρόα ) |
skin |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Parsing Greek Sentences. Parse each word of the sentence found below. For nouns and pronouns, give case and function. For verbs, give person, number, tense, mood, and voice. For adverbs and conjunctions, identify them. For prepositional phrases, give the preposition and the preposition’s object. For adjectives, tell what noun they agree with in gender, number, and case.
καὶ νεβρίδα ἐξάπτω χροὸς θύρσον τε δίδωμι ἐς χεῖρα , κί σσινον βέλος .
Check your answers with those in the Answer Key .
Classics and Our Modern World: Tom Palaima
In his article “Songs of the ‘Hard Traveler’ from Odysseus to the Never-Ending Tourist,” Classicist and professor Tom Palaima studies themes connected with traveling and existing away from home from the Iliad and Odyssey of Homer through the modern folk song tradition as performed and transformed by Bob Dylan, including songs by the Stanley Brothers, Charley Patton, Skip James, Muddy Waters, Stephen F. Foster, Martin Carthy and Dionysis Savvopoulos. Ancient Greek serves as the first recorded examples of songs exploring these experiences.
To read the article, follow this link:
Songs of the Hard Traveler . 1
Module 16 Top 250 Vocabulary to be Memorized. Like learning the alphabet and endings, memorizing vocabulary is essential to acquiring language. The better you memorize the top 250 most frequently occurring words, the greater mastery of the language you will have.
Nouns
ἄνθρωπος , ἀνθρώπου ἡ ὁ human, person
ἔργον , ἔργου τό deed, task, work; building ; ἔργον in truth, in deed
θεός , θεοῦ ἡ ὁ god, goddess, deity
λόγος , λόγου ὁ word, speech, story; reason, account
νόμος , νόμου ὁ law, custom
πόλεμος , πολέμου ὁ war
χρόνος , χρόνου ὁ time
Verb
νομίζω believe, think, have the custom of, hold as custom


8: Retail Human Resources Last updated Save as PDF Page ID 44966 8.1: Introduction to Attracting the Best Employees 8.2: Recruiting Talent 8.3: Developing Star Employees 8.4: Motivating Employees 8.5: Diverse and Inclusive Workforce 8.6: Introduction to Legal Issues in Human Resource Management 8.7: Laws versus Ethics 8.8: Equal Employment Opportunity Law 8.9: Employee Health and Safety Laws 8.10: Sexual Harassment Laws 8.11: Why It Matters- Retail Human Resources 8.12: Web Retailing Laws 8.13: Putting It Together- Retail Human Resources 8.14: Discussion- Retail Human Resources 8.15: Introduction to Organizational Patterns of a Retail Business 8.16: Organizational Structure and Employee Tasks 8.17: Retailer Structures 8.18: Divisions of a Retail Business 8.19: Introduction to Human Resource Management 8.20: Performance Measures 8.21: Steps of Human Resource Management 8.22: Challenges in Human Resource Management