8.7: Rational Equations
Rational Equations
Rational Equations
When one rational expression is set equal to another rational expression, a
rational equation
results.
Some examples of rational equations are the following (except for number 5):
\(\dfrac{3x}{4} = \dfrac{15}{2}\)
\(\dfrac{x+1}{x-2} = \dfrac{x-7}{x-3}\)
\(\dfrac{5a}{2} = 10\)
\(\dfrac{3}{x} + \dfrac{x-3}{x+1} = \dfrac{6}{5x}\)
\(\dfrac{x-6}{x+1}\) is a rational expression , not a rational equation.
The Logic Behind The Process
It seems most reasonable that an equation without any fractions would be easier to solve than an equation with fractions. Our goal, then, is to convert any rational equation to an equation that contains no fractions. This is easily done.
To develop this method, let’s consider the rational equation
\(\dfrac{1}{6} + \dfrac{x}{4} = \dfrac{17}{12}\)
The LCD is 12. We know that we can multiply both sides of an equation by the same nonzero quantity, so we’ll multiply both sides by the LCD, 12.
\(12(\dfrac{1}{6} + \dfrac{x}{4}) = 12 \cdot \dfrac{17}{12}\)
Now distribute 12 to each term on the left side using the distributive property.
\(12 \cdot \dfrac{1}{6} + 12 \cdot \dfrac{x}{4} = 12 \cdot \dfrac{17}{12}\)
Now divide to eliminate all denominators.
\(\begin{array}{flushleft}
2 \cdot 1 + 3 \cdot x &= 17\\
2 + 3x &= 17
\end{array}\)
Now there are no more fractions, and we can solve this equation using our previous techniques to obtain 5 as the solution.
The Process
We have cleared the equation of fractions by multiplying both sides by the LCD. This development generates the following rule.
To clear an equation of fractions, multiply both sides of the equation by the LCD.
When multiplying both sides of the equation by the LCD, we use the distributive property to distribute the LCD to each term. This means we can simplify the above rule.
To clear an equation of fractions, multiply every term on both sides of the equation by the LCD.
The complete method for solving a rational equation is
1. Determine all the values that must be excluded from consideration by finding the values that will produce zero in the denominator (and thus, division by zero). These excluded values are not in the domain of the equation and are called nondomain values.
2. Clear the equation of fractions by multiplying every term by the LCD.
3. Solve this nonfractional equation for the variable. Check to see if any of these potential solutions are excluded values.
4. Check the solution by substitution.
Extraneous Solutions
Potential solutions that have been excluded because they make an expression undefined (or produce a false statement for an equation) are called extraneous solutions. Extraneous solutions are discarded. If there are no other potential solutions, the equation has no solution.
Sample Set A
Solve the following rational equations.
\(\begin{array}{flushleft}
\dfrac{3x}{4} &= \dfrac{15}{2} & \text{ Since the denominators are constants, there are no excluded values.}\\
&& \text{ No values must be excluded. The LCD is 4. Multiply each term by 4}\\
4 \cdot \dfrac{3x}{4} &= 4 \cdot \dfrac{15}{2}\\
\cancel{4} \cdot \dfrac{3x}{\cancel{4}} &= _{\cancel{4}}^{2} \cdot \dfrac{15}{\cancel{2}}\\
3x &= 2 \cdot 15\\
3x &= 30\\
x &= 10 & 10 \text{ is not an excluded value. Check it as a solution}.
\end{array}\)
Check:
\(\begin{array}{flushleft}
\dfrac{3x}{4} &= \dfrac{15}{2}\\
\dfrac{3(10)}{4} &= \dfrac{15}{2} & \text{ Is this correct? }\\
\dfrac{30}{4} &= \dfrac{15}{2} & \text{ Is this correct? }\\
\dfrac{15}{2} &= \dfrac{15}{2} & \text{ Yes, this is correct }
\end{array}\)
\(\begin{aligned}
\dfrac{4}{x-1} &= \dfrac{2}{x+6} & 1 \text{ and } -6 \text{ are nondomain values. Exclude them from the solution}\\
&& \text{The LCD is } (x-1)(x+6) \text{ Multiply every term by the LCD }\\
(x-1)(x+6) \cdot \dfrac{4}{x-1} &= (x-1)(x+6) \cdot \dfrac{2}{x+6}\\
\cancel{(x-1)}(x+6) \cdot \dfrac{4}{\cancel{x-1}} &= (x-1)\cancel{(x+6)} \cdot \dfrac{2}{\cancel{x+6}}\\
4(x+6) &= 2(x-1) & \text{ Solve this nonfractional equation }\\
4x + 24 &= 2x - 2\\
2x &= -26\\
x &= -13 & -13 \text{ is not an excluded value. Check it as a solution}
\end{aligned}\)
Check:
\(\begin{array}{flushleft}
\dfrac{4}{x-1} &= \dfrac{2}{x+6}\\
\dfrac{4}{-13-1} &= \dfrac{2}{-13 + 6} & \text{ Is this correct?}\\
\dfrac{4}{-14} &= \dfrac{2}{-7} & \text{ Is this correct?}\\
\dfrac{2}{-7} &= \dfrac{2}{-13 + 6} & \text{ Yes, this is correct }\\
\end{array}\)
\(-13\) is the solution.
\(\begin{array}{flushleft}
\dfrac{4a}{a-4} &= 2 + \dfrac{16}{a-4}. & 4 \text{ is a nondomain value. Exclude it from consideration}\\
&& \text{ The LCD is } a-4 \text{. Multiply every term by } a-4\\
(a-4) \cdot \dfrac{4a}{a-4} &= 2(a-4) + (a-4) \cdot \dfrac{16}{a-4}\\
\cancel{(a-4)} \cdot \dfrac{4a}{\cancel{a-4}} &= 2(a-4) + \cancel{(a-4)} \cdot \dfrac{16}{\cancel{a-4}}\\
4a &= 2(a-4) + 16 & \text{ Solve this nonfractional equation }\\
4a &= 2a - 8 + 16\\
4a &= 2a + 8\\
2a &= 8\\
a &= 4
\end{array}\)
This value, \(a = 4\), has been excluded from consideration. It is not to be considered as a solution. It is extraneous. As there are no other potential solutions to consider, we conclude that this equation has no solution.
Practice Set A
Solve the following rational equations.
\(\dfrac{2x}{5} = \dfrac{x-14}{6}\)
- Answer
-
\(x=−10\)
\(\dfrac{3a}{a-1} = \dfrac{3a + 8}\)
- Answer
-
\(a=−2\)
\(\dfrac{3}{y-3} + 2 = \dfrac{y}{y-3}\)
- Answer
-
\(y = 3\) is extraneous, so no solution.
Sample Set B
Solve the following rational equations.
\(\begin{array}{flushleft}
\dfrac{3}{x} + \dfrac{4x}{x-1} &= \dfrac{4x^2 + x + 5}{x^2 - x} & \text{ Factor all denominators to find any excluded values and the LCD }\\
&& \text{ Nondomain values are } 0 \text{ and } 1. \text{ Exclude them from consideration. }\\
\dfrac{3}{x} + \dfrac{4x}{x-1} &= \dfrac{4x^2 + x + 5}{x(x-1)} & \text{ The LCD is } x(x-1) \text{. Multiply each term by } x(x-1) \text{ and simplify }
\end{array}\)
\(\cancel{x}(x-1) \cdot \dfrac{3}{\cancel{x}} + x(\cancel{x-1}) \cdot \dfrac{4x}{\cancel{x-1}} = \cancel{x(x-1)} \cdot \dfrac{4x^2 + x + 5}{\cancel{x(x-1)}}\).
\(\begin{array}{flushleft}
3(x-1) + 4x \cdot x &= 4x^2 + x + 5 & \text{ Solve this nonfractional equation to obtain the potential solutions }\\
3x - 3 + 4x^2 &= 4x^2 + x + 5\\
3x - 3 &= x + 5\\
2x &= 8\\
x &= 4 & 4 \text{ is not an excluded value. Check it as a solution }
\end{array}\)
Check:
\(\begin{array}{flushleft}
\dfrac{3}{x} + \dfrac{4x}{x-1} &= \dfrac{4x^2 + x + 5}{x^2 - x}\\
\dfrac{3}{4} + \dfrac{4 \cdot 4}{4-1} &= \dfrac{4 \cdot 4^2 + 4 + 5}{16 - 4} & \text{ Is this correct? }\\
\dfrac{3}{4} + \dfrac{16}{3} &= \dfrac{64 + 4 + 5}{12} & \text{ Is this correct? }\\
\dfrac{9}{12} + \dfrac{64}{12} &= \dfrac{73}{12} & \text{ Is this correct? }\\
\dfrac{73}{12} &= \dfrac{73}{12} & \text{ Yes, this is correct }
\end{array}\)
\(4\) is the solution.
The zero-factor property can be used to solve certain types of rational equations. We studied the zero-factor property in Section 5.1, and you may remember that it states that if \(a\) and \(b\) are real numbers and that \(a \cdot b=0\), then either or both \(a=0\) or \(b=0\).The zero-factor property is useful in solving the following rational equation.
\(\begin{array}{flushleft}
\dfrac{3}{a^2} - \dfrac{2}{a} &= 1 & \text{ Zero is an excluded value. }\\
&& \text{ The LCD is } a^2 \text{ Multiply each term by } a^2 \text{ and simplify }\\
\cancel{a^2} \cdot \dfrac{3}{\cancel{a^2}} - \cancel{a^2} \cdot \dfrac{2}{\cancel{a}} &= 1 \cdot a^2\\
3-2a &= a^2 & \text{ Solve this nonfractional quadratic equation. Set it equal to zero }\\
0 &= a^2 + 2a - 3\\
0 &= (a+3)(a-1)\\
a&= - 3, a = 1 & \text{ Check these as solutions }
\end{array}\)
Check:
\(\begin{array}{flushleft}
\text{If } a = -3: & \dfrac{3}{(-3)^2} - \dfrac{2}{-3} &= 1 & \text{ Is this correct? }\\
& \dfrac{3}{9} + \dfrac{2}{3} &= 1 & \text{ Is this correct? }\\
& \dfrac{1}{3} + \dfrac{2}{3} &= 1 & \text{ Is this correct? }\\
& 1 &= 1 & \text{ Yes, this is correct }\\
& a &= -3 & \text{ Checks and is a solution }\\
\text{If } a = 1: & \dfrac{3}{(1)^2} - \dfrac{2}{1} &= 1 & \text{ Is this correct? }\\
& \dfrac{3}{1} - \dfrac{2}{1} &= 1 & \text{ Is this correct? }\\
& 1 &= 1 & \text{ Yes, this is correct. }\\
& a &= 1 & \text{ Checks and is a solution }
\end{array}\)
\(-3\) and \(1\) are the solutions.
Practice Set B
Solve the equation \(\dfrac{a+3}{a-2} = \dfrac{a+1}{a-1}\)
- Answer
-
\(a = \dfrac{1}{3}\)
Solve the equation \(\dfrac{1}{x-1} - \dfrac{1}{x+1} = \dfrac{2x}{x^2 - 1}\)
- Answer
-
This equation has no solution. \(x=1\) is extraneous.
Section 7.6 Exercises
For the following problems, solve the rational equations.
\(\dfrac{32}{x} = \dfrac{16}{3}\)
- Answer
-
\(x = 6\)
\(\dfrac{54}{y} = \dfrac{27}{4}\)
\(\dfrac{8}{y} = \dfrac{2}{3}\)
- Answer
-
\(y=12\)
\(\dfrac{x}{28} = \dfrac{3}{7}\)
\(\dfrac{x + 1}{4} = \dfrac{x-3}{2}\)
- Answer
-
\(x = 7\)
\(\dfrac{a + 3}{6} = \dfrac{a - 1}{4}\)
\(\dfrac{y-3}{6} = \dfrac{y + 1}{4}\)
- Answer
-
\(y=−9\)
\(\dfrac{x-7}{8} = \dfrac{x+5}{6}\)
\(\dfrac{a + 6}{9} - \dfrac{a-1}{6} = 0\)
- Answer
-
\(a=15\)
\(\dfrac{y + 11}{4} = \dfrac{y + 8}{10}\)
\(\dfrac{b + 1}{2} + 6 = \dfrac{b- 4}{3}\)
- Answer
-
\(b=−47\)
\(\dfrac{m+3}{2} + 1 = \dfrac{m-4}{5}\)
\(\dfrac{a - 6}{2} + 4 = -1\)
- Answer
-
\(a=−4\)
\(\dfrac{b + 11}{3} + 8 = 6\)
\(\dfrac{y - 1}{y + 2} = \dfrac{y + 3}{y - 2}\)
- Answer
-
\(y = -\dfrac{1}{2}\)
\(\dfrac{x + 2}{x - 6} = \dfrac{x - 1}{x + 2}\)
\(\dfrac{3m + 1}{2m} = \dfrac{4}{3}\)
- Answer
-
\(m=−3\)
\(\dfrac{2k + 7}{3k} = \dfrac{5}{4}\)
\(\dfrac{4}{x + 2} = 1\)
- Answer
-
\(x=2\)
\(\dfrac{-6}{x - 3} = 1\)
\(\dfrac{a}{3} + \dfrac{10 + a}{4} = 6\)
- Answer
-
\(a=6\)
\(\dfrac{k + 17}{5} - \dfrac{k}{2} = 2k\)
\(\dfrac{2b + 1}{3b - 5} = \dfrac{1}{4}\)
- Answer
-
\(b = -\dfrac{9}{5}\)
\(\dfrac{-3a + 4}{2a - 7} = \dfrac{-7}{9}\)
\(\dfrac{x}{x + 3} - \dfrac{x}{x-2} = \dfrac{10}{x^2 + x - 6}\)
- Answer
-
\(x=−2\)
\(\dfrac{3y}{y-1} + \dfrac{2y}{y-6} = \dfrac{5y^2 - 15y + 20}{y^2 - 7y + 6}\)
\(\dfrac{4a}{a+2} - \dfrac{3a}{a-1} = \dfrac{a^2 - 8a - 4}{a^2 + a - 2}\)
- Answer
-
\(a=2\)
\(\dfrac{3a - 7}{a-3} = \dfrac{4a - 10}{a - 3}\)
\(\dfrac{2x - 5}{x - 6} = \dfrac{x+1}{x-6}\)
- Answer
-
No solution; 6 is an excluded value.
\(\dfrac{3}{x + 4} + \dfrac{5}{x + 4} = \dfrac{3}{x - 1}\)
\(\dfrac{2}{y + 2} + \dfrac{8}{y + 2} = \dfrac{9}{y + 3}\)
- Answer
-
\(y=−12\)
\(\dfrac{4}{a^2 + 2a} = \dfrac{3}{a^2 + a - 2}\)
\(\dfrac{2}{b(b+2)} = \dfrac{3}{b^2 + 6b + 8}\)
- Answer
-
\(b=8\)
\(\dfrac{x}{x-1} + \dfrac{3x}{x-4} = \dfrac{4x^2 - 8x + 1}{x^2 - 5x + 4}\)
\(\dfrac{4x}{x+2} - \dfrac{x}{x+1} = \dfrac{3x^2 + 4x + 4}{x^2 + 3x + 2}\)
- Answer
-
no solution
\(\dfrac{2}{a-5} - \dfrac{4a - 2}{a^2 - 6a + 5} = \dfrac{-3}{a-1}\)
\(\dfrac{-1}{x+4} - \dfrac{2}{x+1} = \dfrac{4x + 19}{x^2 + 5x + 4}\)
- Answer
-
No solution; \(−4\) is an excluded value.
\(\dfrac{2}{x^2} + \dfrac{1}{x} = 1\)
\(\dfrac{6}{y^2} - \dfrac{5}{y} = 1\)
- Answer
-
\(y=−6, 1\)
\(\dfrac{12}{a^2} - \dfrac{4}{a} = 1\)
\(\dfrac{20}{x^2} - \dfrac{1}{x} = 1\)
- Answer
-
\(x=4, −5\)
\(\dfrac{12}{y} + \dfrac{12}{y^2} = -3\)
\(\dfrac{16}{b^2} + \dfrac{12}{b} = 4\)
- Answer
-
\(y=4,−1\)
\(\dfrac{1}{x^2} = 1\)
\(\dfrac{16}{y^2} = 1\)
- Answer
-
\(y=4,−4\)
\(\dfrac{25}{a^2} = 1\)
\(\dfrac{36}{y^2} = 1\)
- Answer
-
\(y=6,−6\)
\(\dfrac{2}{x^2} + \dfrac{3}{x} = 2\)
\(\dfrac{2}{a^2} - \dfrac{5}{a} = 3\)
- Answer
-
\(a = \dfrac{1}{3}, -2\)
\(\dfrac{2}{x^2} + \dfrac{7}{x} = -6\)
\(\dfrac{4}{a^2} + \dfrac{9}{a} = 9\)
- Answer
-
\(a = -\dfrac{1}{3}, \dfrac{4}{3}\)
\(\dfrac{2}{x} = \dfrac{3}{x+2} + 1\)
\(\dfrac{1}{x} = \dfrac{2}{x+4} - \dfrac{3}{2}\)
- Answer
-
\(x = -\dfrac{4}{3}, -2\)
\(\dfrac{4}{m} - \dfrac{5}{m-3} = 7\)
\(\dfrac{6}{a + 1} - \dfrac{2}{a-2} = 5\)
- Answer
-
\(a = \dfrac{4}{5}, 1\)
For the following problems, solve each literal equation for the designated letter.
\(V = \dfrac{GMm}{D}\) for \(D\)
\(PV = nrt\) for \(n\).
- Answer
-
\(n = \dfrac{PV}{rt}\)
\(E = mc^2\) for \(m\)
\(P = 2(1 + w)\) for \(w\).
- Answer
-
\(W = \dfrac{P - 2}{2}\)
\(A = \dfrac{1}{2}h(b + B)\) for \(B\).
\(A = P(1 + rt)\) for \(r\).
- Answer
-
\(r = \dfrac{A - P}{Pt}\)
\(z = \dfrac{x-\hat{x}}{s}\) for \(\hat{x}\)
\(F=\dfrac{S_{x}^{2}}{S_{y}^{2}} \text { for } S_{y}^{2}\)
- Answer
-
\(S_{y}^{2}=\dfrac{S_{x}^{2}}{F}\)
\(\dfrac{1}{R} = \dfrac{1}{E} + \dfrac{1}{F}\) for \(F\).
\(K = \dfrac{1}{2}h(s_1 + s_2)\) for \(s_2\).
- Answer
-
\(S_{2}=\dfrac{2 K}{h}-S_{1} \text { or } \dfrac{2 K-h S_{1}}{h}\)
\(Q = \dfrac{2mn}{s + t}\) for \(s\).
\(V = \dfrac{1}{6}\pi(3a^2 + h^2)\) for \(h^2\).
- Answer
-
\(h_{2}=\dfrac{6 V-3 \pi a^{2}}{\pi}\)
\(I = \dfrac{E}{R + r}\) for \(R\).
Exercises For Review
Write \((4x^3y^{-4})^{-2}\) so that only positive exponents appear.
- Answer
-
\(\dfrac{y^8}{16x^6}\)
Factor \(x^4 - 16\)
Supply the missing word. An slope of a line is a measure of the _____ of the line.
- Answer
-
steepness
Find the product \(\dfrac{x^{2}-3 x+2}{x^{2}-x-12} \cdot \dfrac{x^{2}+6 x+9}{x^{2}+x-2} \cdot \dfrac{x^{2}-6 x+8}{x^{2}+x-6}\)
Find the sum. \(\dfrac{2x}{x+1} + \dfrac{1}{x-3}\)
- Answer
-
\(\dfrac{2x^2 - 5x + 1}{(x+1)(x-3)}\)


3.4: Brownian Motion on a Phylogenetic Tree
We can use the basic properties of Brownian motion model to figure out what will happen when characters evolve under this model on the branches of a phylogenetic tree. First, consider evolution along a single branch with length t 1 (Figure 3.4A). In this case, we can model simple Brownian motion over time t 1 and denote the starting value as \(\bar{z}(0)\). If we evolve with some rate parameter σ B 2 , then:
\[ E[\bar{z}(t)] \sim N(\bar{z}(0), \sigma_B^2 t_1) \label{3.17} \]
Now consider a small section of a phylogenetic tree including two species and an ancestral stem branch (Figure 3.4B). Assume a character evolves on that tree under Brownian motion, again with starting value \(\bar{z}(0)\) and rate parameter σ B 2 . First consider species a. The mean trait in that species \(\bar{x}_a\) evolves under Brownian motion from the ancestor to species a over a total time of t 1 + t 2 . Thus,
\[ \bar{x}_a \sim N[\bar{z}(0), \sigma_B^2 (t_1+t_2)] \label{3.18}\]
Similarly for species b, over a total time of t 1 + t 3
\[ \bar{x}_b \sim N[\bar{z}(0),\sigma_B^2 (t_1+t_3)] \label{3.19} \]
However, \( \bar{x}_a\) and \( \bar{x}_b\) are not independent of each other. Instead, the two species share one branch in common (branch 1). Each tip trait value can be thought of as an ancestral value plus the sum of two evolutionary changes: one (from branch 1) that is shared between the two species and one that is unique (branch 2 for species a and branch 3 for species b). In this case, mean trait values \( \bar{x}_a\) and \( \bar{x}_b\) will share similarity due to their shared evolutionary history. We can describe this similarity by calculating the covariance between the traits of species a and b. We note that:
\[ \begin{array}{lcr} \bar{x}_a = \Delta \bar{x}_1 + \Delta \bar{x}_2\\ \bar{x}_b = \Delta \bar{x}_1 + \Delta \bar{x}_3\\ \end{array} \label{3.20} \]
Where \(\Delta \bar{x}_1\), \(\Delta \bar{x}_2\), and \(\Delta \bar{x}_3\) represent evolution along the three branches in the tree, are all normally distributed with mean zero and variances σ 2 t 1 , σ 2 t 2 , and σ 2 t 3 , respectively. \(\bar{x}_a\) and \(\bar{x}_b\) are sums of normal random variables and are themselves normal. The covariance of these two terms is simply the variance of their shared term:
\[ cov(\bar{x}_a,\bar{x}_b)=var(\Delta \bar{x}_1)=\sigma_B^2 t_1 \label{3.21} \]
It is also worth noting that we can describe the trait values for the two species as a single draw from a multivariate normal distribution. Each trait has the same expected value, \(\bar{z}(0)\), and the two traits have a variance-covariance matrix:
\[ \begin{bmatrix} \sigma^2 (t_1 + t_2) & \sigma^2 t_1 \\ \sigma^2 t_1 & \sigma^2 (t_1 + t_3) \\ \end{bmatrix} = \sigma^2 \begin{bmatrix} t_1 + t_2 & t_1 \\ t_1 & t_1 + t_3 \\ \end{bmatrix} = \sigma^2 \mathbf{C} \label{3.22} \]
The matrix C in Equation \ref{3.22} is commonly encountered in comparative biology, and will come up again in this book. We will call this matrix the phylogenetic variance-covariance matrix. This matrix has a special structure. For phylogenetic trees with n species, this is an n × n matrix, with each row and column corresponding to one of the n taxa in the tree. Along the diagonal are the total distances of each taxon from the root of the tree, while the off-diagonal elements are the total branch lengths shared by particular pairs of taxa. For example, C (1, 2) and C (2, 1) – which are equal because the matrix C is always symmetric – is the shared phylogenetic path length between the species in the first row – here, species a - and the species in the second row – here, species b. Under Brownian motion, these shared path lengths are proportional to the phylogenetic covariances of trait values. A full example of a phylogenetic variance-covariance matrix for a small tree is shown in Figure 3.5. This multivariate normal distribution completely describes the expected statistical distribution of traits on the tips of a phylogenetic tree if the traits evolve according to a Brownian motion model.


6: Microbial Metabolism (Part A)
Throughout earth’s history, microbial metabolism has been a driving force behind the development and maintenance of the planet’s biosphere. Eukaryotic organisms such as plants and animals typically depend on organic molecules for energy, growth, and reproduction. Prokaryotes, on the other hand, can metabolize a wide range of organic as well as inorganic matter, from complex organic molecules like cellulose to inorganic molecules and ions such as atmospheric nitrogen (N 2 ), molecular hydrogen (H 2 ), sulfide (S 2− ), manganese (II) ions (Mn 2+ ), ferrous iron (Fe 2+ ), and ferric iron (Fe 3+ ), to name a few. By metabolizing such substances, microbes chemically convert them to other forms. In some cases, microbial metabolism produces chemicals that can be harmful to other organisms; in others, it produces substances that are essential to the metabolism and survival of other life forms (Figure \(\PageIndex{1}\)).
-
- 6.1: Energy, Matter, and Enzymes
- Cellular processes such as the building or breaking down of complex molecules occur through series of stepwise, interconnected chemical reactions called metabolic pathways. The term anabolism refers to those endergonic metabolic pathways involved in biosynthesis, converting simple molecular building blocks into more complex molecules, and fueled by the use of cellular energy.
-
- 6.2: Catabolism of Carbohydrates
- Glycolysis is the first step in the breakdown of glucose, resulting in the formation of ATP, which is produced by substrate-level phosphorylation; NADH; and two pyruvate molecules. Glycolysis does not use oxygen and is not oxygen dependent. After glycolysis, a three-carbon pyruvate is decarboxylated to form a two-carbon acetyl group, coupled with the formation of NADH. The acetyl group is attached to a large carrier compound called coenzyme A.
-
- 6.3: Cellular Respiration
- Cellular respiration begins when electrons are transferred from NADH and FADH₂—through a series of chemical reactions to a final inorganic electron acceptor (either oxygen in aerobic respiration or non-oxygen inorganic molecules in anaerobic respiration). These electron transfers take place on the inner part of the cell membrane of prokaryotic cells or in specialized protein complexes in the inner membrane of the mitochondria of eukaryotic cells.
Thumbnail: The Krebs cycle, also known as the citric acid cycle, is summarized here. Note incoming two-carbon acetyl results in the main outputs per turn of two CO 2 , three NADH, one FADH 2 , and one ATP (or GTP) molecules made by substrate-level phosphorylation. Two turns of the Krebs cycle are required to process all of the carbon from one glucose molecule.


1.5: Aperiodic crystal
A periodic crystal is a structure with, ideally, sharp diffraction peaks on the positions of a reciprocal lattice . The structure then is invariant under the translations of the direct lattice . Periodicity here means lattice periodicity . Any structure without this property is aperiodic . For example, an amorphous system is aperiodic. An aperiodic crystal is a structure with sharp diffraction peaks, but without lattice periodicity. Therefore, amorphous systems are not aperiodic crystals. The positions of the sharp diffraction peaks of an aperiodic crystal belong to a vector module of finite rank. This means that the diffraction wave vectors are of the form
\[\mathbf{k}=\sum_{i-1}^{n}h_i\mathbf{a}_i^*, (integer\,h_i) \nonumber \]
The basis vectors \(a_i^*\) are supposed to be independent over the rational numbers, i.e. when a linear combination of them with rational coefficients is zero, all coefficients are zero. The minimum number of basis vectors is the rank of the vector module. If the rank n is larger than the space dimension, the structure is not periodic, but aperiodic.


7.5: Activity 13- Accessible Tree Navigation
Based on the
Tree Menu details
on the previous page, apply what you have learned to the associated activity files to make the tree menu there accessible.
Files for this activity include:
Use the surroundings of the highlighted code on the previous page as a guide to find where the fixes should be applied. Repair the accessibility of the tree menu by applying the highlighted code to the / assets/ik_treemenu.js file.
Note: While we suggest using the highlighted code we’ve provided, you are free to come up with your own solutions provided they produce the expected results listed in the marking rubric below.
Test your updated tree menu with ChromeVox to ensure each element described in the marking rubric below is functioning as suggested.
Requirements
When you have applied your changes and tested to be sure your tree menu functions as described, submit the URL to your tree.html file on your GitHub Pages site, to the file on the web server you are using to host your copy of the activity files, or a GitHack URL.
Grading Rubric
| Criteria | Points |
|---|---|
|
Instructions Provided:
When the tree menu receives focus, instructions are announced on how to use the menu with a keyboard. |
1.0 pts |
|
Tree Menu Semantics:
When navigating through the tree menu with a keyboard, elements are announced with tree menu semantics. |
2.0 pts |
|
Tree Submenus:
When a tree menu item with a submenu receive focus, the submenu state is announced as expanded when open or collapsed when closed. |
2.0 pts |
|
Focus Control:
Only elements of the tree menu that are in view are able to receive focus. |
2.0 pts |
|
Keyboard Operable:
Tree menu functions with a keyboard as described in Adding Keyboard Operability for tree menus. |
3.0 pts |
| Total Points: | 10.0 |


3.9: Culture (2) الثقافة
In this section, you will learn about:
- Months of the year in the Arab world (Hijri / Lunar Calendar).
- Some of the cultural celebrations related to the Hijri Calendar in the Arab World.
Hijri Months:
Muslims around the world follow another calendar, in addition to the Gregorian one, called the Hijri or Lunar Calendar. This Lunar calendar consists of 12 months and is based on the sighting of the new moon every month.
|
No. |
Month Pronunciation |
Arabic Name |
|---|---|---|
|
1 |
Muharram | مُحَرَّم |
|
2 |
Safar | صَفَر |
|
3 |
Rabi' Al-Awwal | رَبِيع ٱلْأَوَّل |
|
4 |
Rabi' Al-Thani / Rabi' Al-Aakher |
رَبِيع ٱلثَّانِي / رَبِيع ٱلْآخِر |
|
5 |
Jumada Al-Oula | جُمَادَىٰ ٱلْأُولَىٰ |
|
6 |
Jumada Al-Thaniyah /Jumada Al-Aakhirah |
جُمَادَىٰ ٱلثَّانِيَة / جُمَادَىٰ ٱلْآخِرَة |
|
7 |
Rajab | رَجَب |
|
8 |
Sha'baan | شَعْبَان |
|
9 |
Ramadan | رَمَضَان |
|
10 |
Shawwal | شَوَّال |
|
11 |
Thu Al-Qa'dah | ذُو ٱلْقَعْدَة |
|
12 |
Thu AL-Hijjah | ذُو ٱلْحِجَّة |
Cultural Celebrations Related to the Hijri Calendar in the Arab World:
The Hijri calendar, also known as the Islamic calendar, follows the lunar cycle and is predominantly used in the Arab world and other Muslim-majority nations. The Islamic calendar holds significant cultural celebrations and religious observances tied to its months and events. To give you an idea, here are some of the main cultural celebrations linked to the Hijri calendar in the Arab world:
1. Islamic New Year / Hijri New Year (رأس السنة الهجريّة/Ra's al Sanah al Hijriyah): The onset of the Islamic lunar calendar is signaled by the Islamic New Year. This event holds great religious importance, as it honors the Hijra (migration) of the Prophet Muhammad from Mecca to Medina. However, in many Arab countries, it is also celebrated as a cultural occasion. The day is often marked with prayers, introspection, and socializing with loved ones.
2. Ramadan (رمضان): Ramadan is the ninth month of the Islamic lunar calendar and is regarded as the most sacred month in Islam. During this month, Muslims fast, pray, engage in reflection and spend time with their community. In the Arab world, Ramadan is observed with special pre-dawn meals known as suhoor and breaking the fast at sunset known as iftar, which is often done communally. Additionally, it is a time for increased acts of charity and spiritual devotion.
3. Eid Al Fitr (عيد الفطر): The Festival of Breaking the Fast, also known as Eid Al Fitr, is celebrated upon the completion of the month-long Ramadan fasting period. This occasion is marked by various activities including prayers, feasting, gift-giving, and charitable acts. It is a time of happiness where families and friends gather together to celebrate. Special events and festivities are organized by communities to commemorate this occasion.
4. Islamic Pilgrimage (الحج/Hajj): Although Hajj is mainly a religious pilgrimage, it has cultural significance throughout the Arab world. During the Islamic month of Dhu Al Hijjah, millions of Muslims from all over the world go to Mecca, Saudi Arabia to perform the pilgrimage. Even for those who do not undertake the pilgrimage, Hajj is a time of increased spirituality and contemplation, and is associated with many cultural traditions.
5. Eid Al Adha (عيد الأضحى): Eid Al Adha, which is also referred to as the Festival of Sacrifice, is an important Islamic holiday that is celebrated by Muslims around the world. It honors the willingness of Prophet Ibrahim (known as Abraham in the Judeo-Christian tradition) to sacrifice his son Isma'il (Ishmael) as an act of obedience to God's command. However, before he could carry out the sacrifice, God provided a ram instead. The festival falls on the 10th day of Dhu Al Hijjah, which is the twelfth month of the Islamic lunar calendar. It is observed following the completion of the Hajj pilgrimage in Mecca, Saudi Arabia. The dates of the festival vary each year based on the Islamic lunar calendar.
6. Prophet Muhammad's Birthday (عيد المولد النبويّ/Eid Al Mawlid Al Nabawi): The celebration of Eid Al Mawlid Al Nabawi takes place during the month of Rabi' Al Awwal in the Islamic calendar and is held in honor of the birth of the Prophet Muhammad. This occasion is marked with various activities such as prayers, recitations of poetry, lectures on the life and teachings of the Prophet, and charitable acts. Processions, decorations, and special foods are also part of the celebrations in some Arab countries, particularly in Egypt and Morocco.
In the Arab world, there are various cultural celebrations that are associated with the Islamic calendar. The cultural practices, rituals, and observances in the region are significantly influenced by the Hijri calendar. The Hijri calendar holds a central position in shaping these aspects of culture.
Activities:
Month Names Crosswords:
Choose the correct option:
Choose the correct option:
Choose the correct option:


16.4: Franchises
A franchise is when a business grants to another the sole right of engaging in a certain business or in a business using a particular trademark in a certain area. Franchises are not a separate form of business organization. Rather, they are a form of contract between businesses. Most franchises involve corporations or limited liability corporations, but they may include sole proprietorships and partnerships.
The advantage of a franchise is that under a franchise agreement , an entrepreneur can open and run a business under a proven business model. The local owner, the franchisee , uses the franchisor’s trademark, intellectual property, and business model under a license agreement. The franchisee offers goods or services to the public and keeps any income earned. In exchange for the right to sell goods or services developed by the franchisor, the franchisee pays a fee to the franchisor.
Franchises are common in some industries such as fast food restaurants, hotels, and tax preparation services. Franchise agreements are very detailed and often require the franchisee to use specific vendors, ingredients, store layouts, colors, etc.
Franchises are also very popular with US businesses interested in conducting business abroad. US businesses collect franchise fees from owners in other nations who are responsible for running the business abroad. This allows US companies to have a presence in nations that may restrict business ownership by foreigners because the businesses themselves are owned and operated by locals.


18: Seed Libraries Fight for the Right to Share (Cook)
Christopher D. Cook
#reportinginformation, #sharedvalues, #kairos, #ethos, #logos, #pathos, #descriptive, #currentevents, #seeds #intellectualproperty
It’s easy to take seeds for granted. Tiny dry pods hidden in packets and sacks, they make a brief appearance as gardeners and farmers collect them for future planting then later drop them into soil. They are not “what’s for dinner,” yet without them there would be no dinner. Seeds are the forgotten heroes of food—and of life itself.
Sharing these wellsprings of sustenance may sound innocuous enough, yet this increasingly popular exchange—and wider seed access—is up against a host of legal and economic obstacles. The players in this surreal saga, wherein the mere sharing of seeds is under attack, range from agriculture officials interpreting seed laws, to powerful corporations expanding their proprietary and market control.
Seed libraries—a type of agricultural commons where gardeners and farmers can borrow and share seed varieties, enriching their biodiversity and nutrition—have sprouted up across the U.S. in recent years, as more Americans seek connection to food and the land. This new variety of seed sharing has blossomed from just a dozen libraries in 2010 to more than 300 today. The sharing of seeds “represents embedded knowledge that we’ve collected over 10,000 years,” says Jamie Harvie, executive director of the Institute for a Sustainable Future, based in Duluth, Minnesota. “Healthy resilient communities are characterized not by how we control other people, and more about valuing relationships.”
As Harvie suggests, seed libraries offer a profound alternative to the corporate takeover of seeds, which has reached frightful proportions: according to the non-profit ETC Group, just three firms control more than half of the worldwide seed business (more than doubling their 22% share in 1996), while the top ten corporations now occupy 76 percent of the global market. Monsanto alone has 26 percent of the world’s seed market, with Du Pont and Syngenta not far behind.
A 2013 report by ETC Group shows the startling scope of the industry’s market power, across the panorama of seeds, agrochemicals, and genetics: Four firms control 58.2% of seeds; 61.9% of agrochemicals; 24.3% of fertilizers; 53.4% of animal pharmaceuticals; and, in livestock genetics, 97% of poultry and two-thirds of swine and cattle research.
Kristina Hubbard, communications director for the Organic Seed Alliance, sees a direct connection between corporate control and the seed-sharing movement. “I think community-based projects like seed libraries are at least in part a direct response to concerns people have about who controls our seed,” explains Hubbard. “It’s a necessary response, as seed industry consolidation continues and is increasing the vulnerability of our seed and food systems. We need more decision makers in the form of seed stewards, and more resiliency in our seed and food systems.”
Seed Libraries Rising
“Love the earth around you,” urges Betsy Goodman, a 27-year-old farmer in Western Iowa, where “most of the landscape is covered in uniform rows of corn and soybeans.” Working on an 11-acre organic farm that sprouts 140 varieties of tomatoes and 60 varieties of peppers, among other crops, Goodman has become something of a seed evangelist. In 2012, she launched the Common Soil Seed Library, just across the Missouri River in nearby Omaha, Nebraska—enabling area gardeners and farmers to borrow some 5,000 seed packets (112 different varieties) to date.
“It didn’t make sense to me that no one was perpetuating the cycle of seed and life,” says Goodman. “People have this idea that you put a seed in the ground, harvest your food, and let it die.” Goodman says she is working to perpetuate life. “The basis of our whole food system comes from the seed,” she says. “I think people are not generally conscious of how grateful we should be for our food diversity and wealth.”
Goodman sees the seed library as an essential reclaiming of farming traditions and local food security. “I want farmers to go back to saving seeds. It’s our responsibility to uphold our food system. It takes everybody.” But, she says, many farmers remain isolated and unaware of the seed-sharing movement. “The consciousness around this is not there yet. I haven’t really heard from farmers yet…The farmers buy their seed each year from Monsanto and Syngenta, this huge industrial system that’s very much in control of this state and surrounding states.” Farmers, she adds, “rely on these companies to buy their corn, they are very tied into these companies, and can’t even feed themselves off of the food they’re growing.”
Motivated by similar concerns, the Wisconsin Seed Savers Alliance has helped germinate six seed libraries (with three more on the way this spring) across five counties in the state’s economically isolated northeast, along the shores of Lake Superior.
“A lot of food grown here is shipped away,” says Alliance director Tessah Wickus. She explains that seed libraries are about “sharing the burden of growing food and making sure we all have something nutritious…We don’t have a whole lot of income sources, our schools are in the system for hot lunch programs, and we have a high poverty rate. One of the concerns here is food security and expanding local foods.”
While small in scale, Wisconsin’s seed library alliance has tapped a well of interest among new farmers and old, says Wickus, who is 25. “Sharing seeds is part of helping the next generation of farmers…[T]his is an integral part of how to survive and sustain yourself, how to pass along knowledge from one generation to another. People have a hunger to know where their food comes from, something we’ve lost.”
About 200 miles westward, on the White Earth Indian Reservation in northern Minnesota, a new seed library offers varieties of sunflower, bean, corn and other seeds to residents—many of whom are poor and seeking a reconnection to indigenous food and farming. Most of the money here “goes off the reservation,” says Zachary Paige, farm manager at the White Earth Land Recovery Project. “This is one way to get the economy back on the reservation, and save money for food, instead of buying seeds from catalogs,” he says, while also “closing that loop in producing food.”
Paige (who is not Native American) helped start the White Earth Seed Library two years ago, and is working with local college and school garden projects to cultivate traditional seed varieties. He points to an indigenous tradition of growing and sharing food, and a revival of highly nutritious pre-Columbus crops, such as Bear Island Corn. Sharing seeds fits into a larger goal on the reservation of “trying to eat healthier and relieve diabetes.”
Seed-Sharing Crackdown
But all this seed-sharing love is butting up against some prodigious economic and regulatory challenges. As the libraries spread across the US, they are catching scrutiny from agriculture officials in states such as Pennsylvania, Minnesota, and Iowa, who express concerns about unlabeled seed packets, and the spreading of contaminated seeds and noxious or invasive species.
One flashpoint in this battle is a small seed library in Mechanicsburg, Pennsylvania, which ran into a regulatory dispute with the state’s department of agriculture. Last June, the Pennsylvania Department of Agriculture informed an employee of the Joseph T. Simpson Public Library that its seed library ran afoul of state seed laws and would have to shut down or follow exorbitant testing and labeling rules intended for commercial seed enterprises. County Commissioner Barbara Cross raised the specter of terrorism, telling local media, “Agri-terrorism is a very, very real scenario,” she said. “Protecting and maintaining the food sources of America is an overwhelming challenge...so you’ve got agri-tourism on one side and agri-terrorism on the other.”
The library was forced to limit its sharing, holding a special seed swapping event instead. As Mechanicsburg seed librarian Rebecca Swanger explained to media at the time, “We can only have current-year seeds, which means 2014, and they have to be store-purchased because those seeds have gone through purity and germination rate testing. People can't donate their own seeds because we can't test them as required by the Seed Act.”
While the Sustainable Economies Law Center (SELC) insists that laws regulating large commercial seed companies do not apply to seed libraries, “other states are now considering adopting Pennsylvania’s seed protocol,” Shareable reported—potentially stopping the seed library movement in its tracks.
But Pennsylvania and some other states “have misapplied the law entirely,” says Neil Thapar, staff attorney at SELC, which is spearheading a national seed library campaign called Save Seed Sharing. Pennsylvania’s Seed Act, he says, “does not actually authorize the state agriculture department to regulate noncommercial seed sharing through seed libraries.”
Thapar argues that applying state seed laws to the libraries is “inappropriate because it violates the original spirit and intent of these laws. Seed laws were created solely as consumer protection laws to protect farmers from unscrupulous seed companies in the marketplace.” In contrast, seed sharing takes place outside of markets, as a “noncommercial activity in community.”
Minnesota’s budding seed library movement has encountered similar resistance. Last September, the state’s department of agriculture (MDA) informed the Duluth Seed Library that it was in violation of state seed laws that prohibit transferring ownership of seeds without comprehensive testing. Harvie, who helped organize the library effort in Duluth, recalls the crackdown “really shocked people…it seemed like an egregious overreach.”
Harvie says the Department of Agriculture enforcements nationally are galvanizing people to support seed libraries. “What people are asking is, who’s being hurt,” he says. “Nobody is being hurt. The only one anyone can imagine being hurt is the seed industry.”
Was the seed industry behind the MDA’s actions? Harvie does not suspect a conspiracy, but he notes, “There had to be some pressure, the [MDA] has plenty of other things to do. Perhaps the MDA knew that by purposefully enforcing the law, it would draw out support for saving.”
Minnesota’s Seed Program Advisory Group, which advises the MDA on state seed laws, meets three times a year and publishes no records of its meetings. Its members include major state commodity groups such as the Minnesota Corn Growers Association, Soybean Growers Association, and the Minnesota Crop Improvement Association.
When the advisory group met last December, Harvie recalls, “I think the Department of Agriculture was excited for us to be at the meeting. It provided them with some community voice,” he says, “when too often it is only industry that can afford the time and expense of attending meetings. The lesson is, the community has to stand up and be present.”
With nationwide challenges to seed libraries, activists worry about a chilling effect on this nascent and increasingly popular form of seed-sharing. In Omaha, Nebraska, the community “has responded really well and been very supportive” of Common Soil’s initiative, says Goodman. “We’re not being attacked, we are being supported,” she says, by gardeners and lawmakers interested in putting the libraries on more solid legal ground. But, she adds, “I was approached by others across Nebraska who wanted to open seed libraries, but they were afraid they would put all this work in and get shut down.”
It remains unclear whether the seed industry has played any role in promoting the enforcement push, but this powerful agribusiness sector is vigilant about expanding its control over seeds. As first reported by MintPress News this January, the conservative American Legislative Exchange Council (ALEC) is peddling its “Pre-Emption of Local Agricultural Laws Act”—a law providing “exclusive regulatory power over agricultural seed, flower seed and vegetable seed and products of agriculture seed, flower seed and vegetable seed to the state.” Despite the conservative mantra of “local control,” ALEC’s measure would prohibit local governments from enacting or enforcing measures to “inhibit or prevent the production or use of agricultural seed, flower seed or vegetable seed or products.”
Meanwhile, the American Seed Trade Association advocates for “Strong intellectual property protection,” to keep investment dollars flowing, and to “add value to agriculture and society through new products. Any state legislation that could undermine this simple principle is vigorously opposed.”
Asked for its stance on seed libraries, the US Department of Agriculture (USDA) stated, “We have not received any formal complaints of mislabeled seed being distributed in interstate commerce through these programs (seed libraries).” The Federal Seed Act (FSA) governs “truthful labeling of agricultural and vegetable seed shipped in interstate commerce,” the agency said, adding, “It remains to be seen if any of the seed being obtained from these libraries will make it into interstate commerce.” Unless the seeds are shipped across state lines, or “determined to be a variety protected” under the federal Seed Act, the FSA “has no jurisdiction over this seed. Individual States will need to establish internal methods of dealing with labeling and possible mislabeling of the seed packets.”
Saving the Libraries
As state agriculture agencies consider whether to curtail seed libraries, legislative efforts are underway in Nebraska, Minnesota, and other states, to protect them. The Community Gardens Act [pdf] currently moving through the Nebraska legislature would exempt seed libraries from state laws governing seed labeling and testing. In December 2014, the city council of Duluth, Minnesota passed a resolution supporting seed sharing “without legal barriers of labeling fees and germination testing.”
Perhaps more significantly, the Duluth resolution advocated reforming the Minnesota Seed Law to “support the sharing of seeds by individuals and through seed libraries,” by exempting these forms of sharing from the law’s labeling, testing, and permitting requirements. After one reform measure was withdrawn from the Minnesota legislature, activists are gearing up for another legislative push soon.
In coming months, seed-sharing advocates can expect legislative battles across the US—some seeking to expand libraries’ sharing rights, and others limiting the exchange. Meanwhile, agribusiness continues to widen its economic and legal control over the world’s seed supply. “Seed sharing is an interactive and vibrant contrast to the extractive marketplace,” says Harvie. The battle over seed libraries and sharing represents “a clash of worldviews that just don’t reconcile.”
Despite the challenges, Goodman remains buoyant about the seed library movement. “It’s natural for companies to try to get power over this, but it’s our responsibility to push back and establish our freedom,” she says. “We are losing huge chunks of our food system, and it’s our responsibility to reclaim it. We have to be the ones to do it.”
____________________
Christopher D. Cook is an award-winning journalist and author of Diet for a Dead Planet: Big Business and the Coming Food Crisis . This essay originally appeared in Shareable.


3.3: Discussion 2- Plate Tectonics
Discussion 2
Plate Tectonics
Introduction
This week we will be having a conversation about plate tectonics and the ways in which the hazards associated with movement of the plates might personally affect you or your family. Now that you have completed Module 3 (Plate Tectonics), you should have some basic knowledge about the subject and can make educated post(s).
For this discussion, you will write an intial post and reply to at least TWO of your classmates’ posts. Follow the guidelines for each posting below:
Basic Guidelines for Initial Post
(20 points, required)
Under the Discussion 2 topic, create a new thread. Answer the following questions thoroughly. Each question should be addressed in complete sentences and 4 – 5 sentences each.
- Where did you grow up, or where do you call home? Where are the closest plate boundaries to your home (name the specific plates involved)? Is the boundary you identified a divergent, transform, or convergent boundary, or some combination of these? Describe this boundary in more detail if possible.
- If a family member asked you to describe the potential danger of living near the plate boundary located closest to your home, what would you tell them?
- Knowing what you do now about plate tectonics and the hazards associated with plate boundaries, how would this new knowledge play a role in where you choose to live in the future? How much risk might you be willing to live with?
- Find two web pages or articles from two different sources that you feel provide a good overview of plate tectonics. How might you determine if it is a credible source for information? Describe what you liked (or didn’t like) about the way the information was presented.
Basic Guidelines for Replies to Students
(10 points, required)
-
Make
two
replies to
two
different students, thoughtfully addressing
- their concerns and/or assessments about the hazards associated with plate tectonics as they relate to them, and
- at least one of the websites they chose as a good source for information on plate tectonics.
- Each reply must contain 6 or 7 meaningful sentences directly related to article content. Phrases like “that is a great article!” and “I really liked your summary” do not count as meaningful content. However, these are still encouraging phrases in support of your peers so please feel free to include them in your post.
- You must reply to at least 2 other students. (3, or more, for extra credit.)
- Do not wait until the last day of the discussion to make your replies. You only receive credit for one reply (up to 10 points max) on the last day.
- Reply early and often, and earning your full discussion points will be easy!
Extra Credit
(5 points, optional)
Make a third reply (or more!) to a student following the guidelines above.
Learning Outcomes
As a result of your participation in the discussion, you will become more aware of current events and news articles relating to topics in geology (i.e. become a more informed citizen). You will have some freedom in customizing this course to pursue your own interests in geology, including things not covered in the content of this course, by finding news articles on topics that relate to you.
Grading Rubric
Initial Post
20 points : Initial post met length and content requirements
15 points : Initial post needs minor improvements in length and content
10 points : Initial post needs major improvements in length or content
5 points : Initial post was made but with very little effort
0 points : Initial post was not made
Reply posts (each)
5 points : Reply was thoughtful, relevant, and met length requirements.
4 points : Reply needs minor improvements in relevancy and/or length.
2.5 points : Reply needs major improvements in relevancy and/or length.
1 points : Reply was made but with very little effort.
0 points : Reply was not made.


3.10: Nomenclature of Ionic Compounds
The simplest ionic compounds consist of a single type of cation associated with a single type of anion. Nomenclature for these compounds is trivial; the cation is named first, followed by the anion. If the anion is a single element, the suffix ide is added to the root name of the element.
When you are constructing names for ionic compounds, you do not use “multipliers” to indicate how many cations or anions are present in the compound. For example NaI is named sodium iodide; Na 2 S is named sodium sulfide; CaCl 2 is named calcium chloride. The chemist reading the name is assumed to have sufficient knowledge to pair the elements properly based on their common valence states. There are exceptions to this simple nomenclature, however. Many transition metals exist as more than one type of cation. Thus, iron exists as Fe 2+ and Fe 3+ cations (they are referred to as “oxidation states”, and will be covered in detail in Chapter 5 ). When you are naming an ionic compound containing iron, it is necessary to indicate which oxidation state the metal has. For metals, the oxidation state is the same as the charge. Thus Fe 2+ in a compound with chloride would have a formula FeCl 2 and would be named iron (II) chloride, with the oxidation state (the charge on the iron) appearing as a Roman numeral in parenthesis after the cation. The cation Fe 3+ paired with oxygen would have the formula Fe 2 O 3 and would have the name iron (III) oxide.
The procedure for naming ionic compounds contain polyatomic ions is identical to that described above for simple ions. Thus, CaCO 3 is named calcium carbonate; Na 2 SO 4 is named sodium sulfate; (NH 4 ) 2 HPO 4 (a compound with two polyatomic ions) is named ammonium hydrogen phosphate; and Pb 2+ paired with SO 4 2- , PbSO 4 is named lead (II) sulfate.
Write a correct chemical formula for each of the following ionic compounds:
- Calcium bromide
- Aluminum oxide
- Copper (II) chloride
- Iron (III) oxide
Solution
- Calcium is 2+, bromide is 1-; CaBr 2 .
- Aluminum is 3+, oxide is 2-; Al 2 O 3 .
- From the oxidation state that is given, copper is 2+, chloride is 1-; CuCl 2 .
- From the oxidation state, iron is 3+, oxide is 2-; Fe 2 O 3 .
Write a proper chemical name for each of the following ionic compounds:
- Li 2 S
- CaO
- NiCl 2
- FeO
Solution
- We don’t use multipliers, so this is simply lithium sulfide.
- This is simply calcium oxide.
- We don’t have to specify an oxidation state for nickel, so this is nickel chloride.
- We must specify that iron is 2+ in this compound; iron (II) oxide.
Write a correct chemical formula for each of the following ionic compounds:
- Sodium phosphide
- Iron (II) nitrite
- Calcium hydrogen phosphate
- Chromium (III) oxide
Write a proper chemical name for each of the following ionic compounds:
- NaBr
- CuCl 2
- Fe(NO 3 ) 3
- (NH 4 ) 3 PO 4


11.3: Elimination by Substitution
When Substitution Works Best
We know how to solve a linear equation in one variable. We shall now study a method for solving a system of two linear equations in two variables by transforming the two equations in two variables into one equation in one variable.
To make this transformation, we need to eliminate one equation and one variable. We can make this
elimination by substitution
.
When Substitution Works Best
The substitution method works best when either of these conditions exists:
- One of the variables has a coefficient of 1, or
- One of the variables can be made to have a coefficient of 1 without introducing fractions.
The Substitution Method
To solve a system of two linear equations in two variables,
- Solve one of the equations for one of the variables.
- Substitute the expression for the variable chosen in step 1 into the other equation.
- Solve the resulting equation in one variable.
- Substitute the value obtained in step 3 into the equation obtained in step 1 and solve to obtain the value of the other variable.
- Check the solution in both equations.
- Write the solution as an ordered pair.
Sample Set A
Solve the system
\(\left\{\begin{array}{r}
2x + 3y = 14 \\
3x + y = 7
\end{array}\right.\)
Step 1: Since the coefficient of \(y\) in equation 2 is \(1\), we will solve equation 2 for \(y\).
\(y = -3x + 7\)
Step 2: Substitute the expression \(-3x + 7\) for \(y\) in equation 1.
\(2x + 3(-3x + 7) = 14\)
Step 3: Solve the equation obtained in step 2.
\(\begin{aligned}
2x + 3(-3x + 7) &= 14\\
2x - 9x + 21 &= 14\\
-7x + 21 &= 14\\
-7x &= -7\\
x &= 1
\end{aligned}\)
Step 4: Substitute \(x = 1\) into the equation obtained in step 1, \(y = -3x + 7\).
\(\begin{aligned}
y &= -3(1) + 7\\
y &= -3 + 7\\
y &= 4
\end{aligned}\)
Step 5: Substitute \(x = 1, y = 4\) into each of the original equations for a check.
1)
\(\begin{aligned}
2 x+3 y &=14 \\
2(1)+3(4) &=14 \text { Is this correct? } \\
2+12 &=14 \text { Is this correct? } \\
14 &=14 \text { Yes, this is correct. }
\end{aligned}\)
2)
\(\begin{aligned}
3x + y &= 7\\
3(1) + (4) &= 7 \text{ Is this correct? }\\
3 + 4 &= 7 \text{ Is this correct? }\\
7 &= 7 \text{ Yes, this is correct. }
\end{aligned}\)
Step 6: The solution is \((1, 4)\). The point \((1, 4)\) is the point of intersection of the two lines of the system.
Practice Set A
Solve the system
\(\left\{\begin{array}{r}
5x - 8y = 18 \\
4x + y = 7
\end{array}\right.\)
- Answer
-
The point \((2, -1)\) is the point of intersection of the two lines.
Substitution And Parallel Lines
The following rule alerts us to the fact that the two lines of a system are parallel.
If computations eliminate all the variables and produce a contradiction, the two lines of a system are parallel, and the system is called inconsistent.
Sample Set B
Solve the system
\(\left\{\begin{array}{r}
2x - y = 1 \\
4x - 2y = 4
\end{array}\right.\)
Step 1: Solve equation for \(y\).
\(\begin{aligned}
2x - y &= 1\\
-y &= -2x + 1\\
y &= 2x - 1
\end{aligned}\)
Step 2: Substitute the expression \(2x - 1\) for \(y\) into equation 2.
\(4x - 2(2x - 1) = 4\)
Step 3: Solve the equation obtained in step 2.
\(\begin{aligned}
4x - 2(2x - 1) &= 4\\
4x - 4x + 2 &= 4\\
2 &\not= 4
\end{aligned}\)
Computations have eliminated all the variables and produce a contradiction. These lines are parallel.
This system is inconsistent.
Practice Set B
Solve the system
\(\left\{\begin{array}{r}
7x - 3y = 2 \\
14x - 6y = 1
\end{array}\right.\)
- Answer
-
Substitution produces \(4 \not= 1\), or \(\dfrac{1}{2} \not = 2\), a contradiction. These lines are parallel and the system is inconsistent.
Substitution And Coincident Lines
The following rule alerts us to the fact that the two lines of a system are coincident.
If computations eliminate all the variables and produce an identity, the two lines of a system are coincident and the system is called dependent.
Sample Set C
Solve the system
\(\left\{\begin{array}{r}
4x + 8y = 8 \\
3x + 6y = 6
\end{array}\right.\)
Step 1: Divide equation 1 by \(4\) and solve for \(x\).
\(\begin{aligned}
4x + 8y &= 8\\
x + 2y &= 2\\
x &= -2y + 2
\end{aligned}\)
Step 2: Substitute the expression \(-2y + 2\) for \(x\) in equation 2.
\(3(-2y + 2) + 6y = 6\)
Step 3: Solve the equation obtained in step 2.
\(\begin{aligned}
3(-2y + 2) + 6y &= 6\\
-6y + 6 + 6y &= 6\\
6 &= 6
\end{aligned}\)
Computations have eliminated all the variables and produced an identity. These lines are coincident.
This system is dependent.
Practice Set C
Solve the system
\(\left\{\begin{array}{r}
4x + 3y = 1 \\
-8x - 6y = -2
\end{array}\right.\)
- Answer
-
Computations produce \(-2 = -2\), an identity. These lines are coincident and the system is dependent.
Systems in which a coefficient of one of the variables is not \(1\) or cannot be made to be \(1\) without introducing fractions are not well suited for the substitution method. The problem in Sample Set D illustrates this “messy” situation.
Sample Set D
Solve the system
\(\left\{\begin{array}{r}
3x + 2y = 1 \\
4x - 3y = 3
\end{array}\right.\)
Step 1: We will solve equation (1) for \(y\).
\(\begin{aligned}
3x + 2y &= 1\\
2y &= -3x + 1\\
y &= \dfrac{-3}{2}x + \dfrac{1}{2}
\end{aligned}\)
Step 2: Substitute the expression \(\dfrac{-3}{2}x + \dfrac{1}{2}\) for \(y\) in equation (2).
\(4x - 3(\dfrac{-3}{2}x + \dfrac{1}{2}) = 3\)
Step 3: Solve the equation obtained in step 2.
\(\begin{aligned}
4x - 3(\dfrac{-3}{2}x + \dfrac{1}{2}) &= 3 \text{ Multiply both sides by the LCD, } 2\\
4x + \dfrac{9}{2}x - \dfrac{3}{2} &= 3\\
8x + 9x - 3 &= 6\\
17x - 3 &= 6\\
17x &= 9\\
x &= \dfrac{9}{17}
\end{aligned}\)
Step 4: Substitute \(x = \dfrac{9}{17}\) into the equation obtained in step 1, \(y = \dfrac{-3}{2}x + \dfrac{1}{2}\)
\(y = \dfrac{-3}{2}(\dfrac{9}{17}) + \dfrac{1}{2}\\
y = \dfrac{-27}{34} + \dfrac{17}{34} = \dfrac{-10}{34} = \dfrac{-5}{17}\).
We now have \(x = \dfrac{9}{17}\) and \(y = \dfrac{-5}{17}\)\
Step 5: Substitution will show that these values of \(x\) and \(y\) check.
Step 6: The solution is \((\dfrac{9}{17}, \dfrac{-5}{17})\)
Practice Set D
Solve the system
\(\left\{\begin{array}{r}
9x - 5y = -4 \\
2x + 7y = -9
\end{array}\right.\)
- Answer
-
These lines intersect at the point \((−1,−1)\).
Exercises
For the following problems, solve the systems by substitution.
\(\left\{\begin{array}{r}
3x + 2y = 9 \\
y = -3x + 6
\end{array}\right.\)
- Answer
-
\((1, 3)\)
\(\left\{\begin{array}{r}
5x - 3y = -6 \\
y = -4x + 12
\end{array}\right.\)
\(\left\{\begin{array}{r}
2x + 2y = 0 \\
x = 3y - 4
\end{array}\right.\)
\(\left\{\begin{array}{r}
9x - 5y = -4 \\
2x + 7y = -9
\end{array}\right.\)
- Answer
-
\((-1, 1)\)
\(\left\{\begin{array}{r}
3x + 5y = 9 \\
x = 4y - 14
\end{array}\right.\)
\(\left\{\begin{array}{r}
-3x + y = -4\\
2x + 3y = 10
\end{array}\right.\)
- Answer
-
\((2,2)\)
\(\left\{\begin{array}{r}
-4x + y = -7 \\
2x + 5y = 9
\end{array}\right.\)
\(\left\{\begin{array}{r}
6x - 6 = 18 \\
x + 3y = 3
\end{array}\right.\)
- Answer
-
\((4, -\dfrac{1}{3})\)
\(\left\{\begin{array}{r}
-x - y = 5 \\
23x + y = 5
\end{array}\right.\)
\(\left\{\begin{array}{r}
-5x + y = 4 \\
10x - 2y = -8
\end{array}\right.\)
- Answer
-
Dependent (same line)
\(\left\{\begin{array}{r}
x + 4y = 1 \\
-3x - 12y = -1
\end{array}\right.\)
\(\left\{\begin{array}{r}
4x - 2y = 8 \\
6x + 3y = 0
\end{array}\right.\)
- Answer
-
\((1,−2)\)
\(\left\{\begin{array}{r}
2x + 3y = 12 \\
2x + 4y = 18
\end{array}\right.\)
\(\left\{\begin{array}{r}
3x - 9y = 6 \\
6x - 18y = 5
\end{array}\right.\)
- Answer
-
inconsistent (parallel lines)
\(\left\{\begin{array}{r}
-x + 4y = 8 \\
3x - 12y = 10
\end{array}\right.\)
\(\left\{\begin{array}{r}
x + y = -6 \\
x - y = 4
\end{array}\right.\)
- Answer
-
\((−1,−5)\)
\(\left\{\begin{array}{r}
2x + y = 0 \\
x - 3y = 0
\end{array}\right.\)
\(\left\{\begin{array}{r}
4x - 2y = 7 \\
y = 4
\end{array}\right.\)
- Answer
-
\((\dfrac{15}{4}, 4)\)
\(\left\{\begin{array}{r}
x + 6y = 11 \\
x = -1
\end{array}\right.\)
\(\left\{\begin{array}{r}
2x - 4y = 10 \\
3x = 5y + 12
\end{array}\right.\)
- Answer
-
\((−1,−3)\)
\(\left\{\begin{array}{r}
y + 7x + 4 = 0 \\
x = -7y + 28
\end{array}\right.\)
\(\left\{\begin{array}{r}
x + 4y = 0 \\
x + \dfrac{2}{3}y = \dfrac{10}{3}
\end{array}\right.\)
- Answer
-
\((4,−1)\)
\(\left\{\begin{array}{r}
x = 24 - 5y \\
x - \dfrac{5}{4}y = \dfrac{3}{2}
\end{array}\right.\)
\(\left\{\begin{array}{r}
x = 11 - 6y \\
3x + 18y = -33
\end{array}\right.\)
- Answer
-
inconsistent (parallel lines)
\(\left\{\begin{array}{r}
2x + \dfrac{1}{3}y = 4 \\
3x + 6y = 39
\end{array}\right.\)
\(\left\{\begin{array}{r}
\dfrac{4}{5}x + \dfrac{1}{2}y = \dfrac{3}{10} \\
\dfrac{1}{3} + \dfrac{1}{2}y = \dfrac{-1}{6}
\end{array}\right.\)
- Answer
-
\((1,−1)\)
Exercises For Review
Find the quotient: \(\dfrac{x^2 - x - 12}{x^2 - 2x - 15} \div \dfrac{x^2 - 3x - 10}{x^2 - 2x - 8}\)
- Answer
-
\(\dfrac{(x-4)^2}{(x-5)^2}\)
Find the difference: \(\dfrac{x + 2}{x^2 + 5x + 6} - \dfrac{x + 1}{x^2 + 4x + 3}\)
Simplify \(-\sqrt{81x^8y^5z^4}\)
- Answer
-
\(-9x^4y^2z^2 \sqrt{y}\)
Use the quadratic formula to solve \(2x^2 + 2x - 3 = 0\)
Solve by graphing:
\(\left\{\begin{array}{r}
x - y = 1 \\
2x + y = 5
\end{array}\right.\)
- Answer
-
\((2,1)\)


4: Probability and Odds
The probability of a specified event is the chance or likelihood that it will occur. There are several ways of viewing probability. One would be experimental in nature, where we repeatedly conduct an experiment. Suppose we flipped a coin over and over and over again and it came up heads about half of the time; we would expect that in the future whenever we flipped the coin, it would turn up heads about half of the time. When a weather reporter says “there is a 10% chance of rain tomorrow,” she is basing that on prior evidence; that out of all days with similar weather patterns, it has rained on 1 out of 10 of those days.
Another view would be subjective in nature, an educated guess. If someone asked you the probability that the Los Angeles Dodgers would win their next baseball game, it would be impossible to conduct an experiment where the same two teams played each other repeatedly, each time with the same starting lineup and starting pitchers, each starting at the same time of day on the same field under the precisely the same conditions. Since there are so many variables to take into account, someone familiar with baseball and with the two teams involved might make an educated guess that there is a 75% chance they will win the game; that is, if the same two teams were to play each other repeatedly under identical conditions, the Dodgers would win about three out of every four games. But this is just a guess, with no way to verify its accuracy, and depending upon how educated the educated guesser is, a subjective probability may not be worth very much.
We will return to the experimental and subjective probabilities from time to time, but in this course we will mostly be concerned with theoretical probability, which is defined as follows: Suppose there is a situation with n equally likely possible outcomes and that m of those n outcomes correspond to a particular event; then the probability of that event is defined as \(m/n\).
-
- 4.3: Counting
- What we'll be investigating here are ways of counting efficiently. When we get to the probability situations a bit later in this chapter we will need to count some very large numbers, like the number of possible winning lottery tickets. One way to do this would be to write down every possible set of numbers that might show up on a lottery ticket, but believe me: you don't want to do this.
-
- 4.4: Expected Value
- Expected value is perhaps the most useful probability concept we will discuss. It has many applications, from insurance policies to making financial decisions, and it's one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0