16.3: Key Financial Statements
What you’ll learn to do: identify key financial statements and their components, and explain the primary use of each type of statement
In this section you will learn about key financial statements of accounting: the balance sheet, income statement, statement of owner’s equity, and statement of cash flows. By examining the components of each you will see the connections between the statements and be able to use this information to help you determine the point at which your business becomes profitable—the break-even point.
Learning Objectives
- Identify the use and components of the balance sheet
- Identify the use and components of the statement of owner’s equity
- Explain how the balance sheet, income statement, statement of owner’s equity, and statement of cash flows are connected
Financial Statements
Financial statements are the means by which companies communicate their story. Together these statements represent the profitability and financial strength of a company. The financial statement that reflects a company’s profitability is the income statement. The statement of owner’s equity —also called the statement of retained earnings —shows the change in retained earnings between the beginning and end of a period (e.g., a month or a year). The balance sheet reflects a company’s solvency and financial position. The statement of cash flows shows the cash inflows and outflows for a company during a period of time.
Financial statements are summative reports in that they report information obtained from the day-to-day bookkeeping activities of financial accountants or bookkeepers. After all of the income and expenses of the business have been recorded, financial accountants prepare financial statements in the following order:
- Income Statement
- Statement of Retained Earnings—also called Statement of Owner’s Equity
- The Balance Sheet
- The Statement of Cash Flows
The following video summarizes the four financial statements required by GAAP.
In order to get a better understanding of financial statements, what they communicate to the users of accounting information, and how the statements are connected, we will use the final balances as of January 31, 20XX for a fictitious delivery-service company, Metro Courier Inc. Just as a financial accountant would do, we will use these figures to prepare the company’s financial statements required by GAAP.
Before we start, we need to define three terms and an equation that are used throughout the accounting process.
- Asset : An asset is an economic resource. Anything tangible or intangible that can be owned or controlled to produce value and that is held to have positive economic value is considered an asset. Simply stated, assets represent value of ownership that can be converted into cash (although cash itself is also considered an asset). Assets include things like cash, vehicles, buildings, equipment, patents, and debts owed to the company.
- Liability : A liability is defined as the future sacrifices of economic benefits that the entity is obliged to make to other entities as a result of past transactions or other past events, the settlement of which may result in the transfer or use of assets, provision of services, or other yielding of economic benefits in the future. Liabilities include things like loans, monies owed to suppliers or creditors that the business will use assets (i.e., cash) to settle.
- Equity : Equity is the difference between the value of the assets and the amount of the liabilities of something owned. Owner’s equity consists of the net assets of an entity. Net assets is the difference between the total assets and total liabilities. When the owners are shareholders, the interest can be called shareholders’ equity; the accounting remains the same, and it is ownership equity spread out among shareholders.
You can see that these three terms are interconnected, and their interconnection produces an equation that is at the heart of all financial accounting: The Accounting Equation . The accounting equation represents the relationship between assets, liabilities, and the owner’s equity of a business. It’s the foundation for the double-entry accounting system, accepted to be the most reliable and accurate method of recording the financial transactions of a business. The accounting equation must always “balance”: The left and right side of the equation must be equal. The accounting equation is as follows:
Now that you have a better understanding of the language of financial statements, let’s look at Metro Courier’s financial information and prepare some financial statements.
| Cash | Asset | $ 66,800 |
| Accounts Receivable | Asset | $ 5,000 |
| Supplies | Asset | $ 500 |
| Prepaid rent | Asset | $ 1,800 |
| Equipment | Asset | $ 5,500 |
| Truck | Asset | $ 8,500 |
| Accounts Payable | Liability | $ 200 |
| Common Stock | Equity | $ 30,000 |
| Retained Earnings | Equity | $ 0 |
| Service Revenue | Revenue | $ 60,000 |
| Salary Expense | Expense | $ 900 |
| Utilities Expense | Expense | $ 1,200 |
Income Statement
The income statement, sometimes called an earnings statement or profit and loss statement, reports the profitability of a business organization for a stated period of time. In accounting, we measure profitability for a period, such as a month or year, by comparing the revenues earned with the expenses incurred to produce these revenues. This is the first financial statement prepared, as you will need the information from this statement for the remaining statements. The income statement contains the following:
- Revenues are the inflows of cash resulting from the sale of products or the rendering of services to customers. We measure revenues by the prices agreed on in the exchanges in which a business delivers goods or renders services.
- Expenses are the costs incurred to produce revenues. Expenses are costs of doing business (typically identified as accounts ending in the word “expense”).
- Revenues – Expenses = Net Income . Net income is often called the earnings of the company. When expenses exceed revenues, the business has a net loss .
| Metro Courier Inc. | ||
| Income Statement | ||
| Month Ended January 31, 20XX | ||
| Revenue: | ||
| Service Revenue | $ 60,000 | |
| Total Revenues | $ 60,000 | |
| Expenses: | ||
| Salary Expense | 900 | |
| Utility Expense | 1, 200 | |
| Total Expenses | 2,100 | |
| Net Income ($60,000 – 2,100) | $ 57,900 |
The net income from the income statement will be used in the Statement of Equity.
Statement of Retained Earnings (or Owner’s Equity)
The statement of retained earnings , explains the changes in retained earnings between two balance sheet dates. We start with beginning retained earnings (in our example, the business began in January, so we start with a zero balance) and add any net income (or subtract net loss) from the income statement. Next, we subtract any dividends declared (or any owner withdrawals in a partnership or sole-proprietor) to get the ending balance in retained earnings (or capital for non-corporations)
| Metro Courier Inc. | |
| Statement of Retained Earnings | |
| Month Ended January 31, 20XX | |
| Beginning Retained Earnings, Jan 1 | $ 0 |
| Net income from month (from income statement) | 57,900 |
| Total increase | $ 57,900 |
| Dividends (or withdrawals for non-corporations) | – $0 |
| Ending Retained Earnings, January 31 | $ 57,900 |
The ending balance we calculated for retained earnings (or capital) is reported on the balance sheet.
Balance Sheet
The balance sheet lists the company’s assets, liabilities, and equity (including dollar amounts) as of a specific moment in time. That specific moment is the close of business on the date of the balance sheet. Notice how the heading of the balance sheet differs from the headings on the income statement and statement of retained earnings. A balance sheet is like a photograph; it captures the financial position of a company at a particular moment in time. The other two statements are for a period of time. As you learn about the assets, liabilities, and stockholders’ equity contained in a balance sheet, you will understand why this financial statement provides information about the solvency of the business.
| Metro Courier Inc. | ||||
| Balance Sheet | ||||
| January 31, 20XX | ||||
| Assets | Liabilities and Equity | |||
| Cash | $ 66,800 | Accounts Payable | 200 | |
| Accounts Receivable | 5,000 | Total Liabilities | 200 | |
| Supplies | 500 | |||
| Prepaid Rent | 1,800 | Common Stock | 30,000 | |
| Equipment | 5,500 | Retained Earnings | 57,900 | |
| Truck | 8,500 | Total Equity | 87,900 | |
| Total Assets | $ 88,100 | Total Liabilities + Equity | $ 88,100 |
You can see the accounting equation in action here on the balance sheet. The accounting equation is Assets – Liabilities = Owner’s Equity. For Metro Courier Inc., this is $88,100 – $200 = $87,900.
Statement of Cash Flows
The main purpose of the statement of cash flows is to report on the cash receipts and cash disbursements of an entity during an accounting period. Broadly defined, cash includes both cash and cash equivalents, such as short-term investments in Treasury bills, commercial paper, and money market funds. Another purpose of this statement is to report on the entity’s investing and financing activities for the period. The statement of cash flows reports the effects on cash during a period of a company’s operating, investing, and financing activities. Firms show the effects of significant investing and financing activities that do not affect cash in a schedule separate from the statement of cash flows.
The statement of cash flows summarizes the effects on cash of the operating, investing, and financing activities of a company during an accounting period; it reports on past management decisions on such matters as issuance of capital stock or the sale of long-term bonds. This information is available only in bits and pieces from the other financial statements. Since cash flows are vital to a company’s financial health, the statement of cash flows provides useful information to management, investors, creditors, and other interested parties.
The statement of cash flows presents the effects on cash of all significant operating, investing, and financing activities. By reviewing the statement, management can see the effects of its past major policy decisions in quantitative form. The statement may show a flow of cash from operating activities large enough to finance all projected capital needs internally rather than having to incur long-term debt or issue additional stock. Alternatively, if the company has been experiencing cash shortages, management can use the statement to determine why such shortages are occurring. Using the statement of cash flows, management may also recommend to the board of directors a reduction in dividends to conserve cash.
The statement of cash flows classifies cash receipts and disbursements as operating, investing, and financing cash flows. Both inflows and outflows are included within each category.
Operating activities generally include the cash effects (inflows and outflows) of transactions and other events that enter into the determination of net income. Cash inflows from operating activities affect items that appear on the income statement and include: (1) cash receipts from sales of goods or services; (2) interest received from making loans; (3) dividends received from investments in equity securities; (4) cash received from the sale of trading securities; and (5) other cash receipts that do not arise from transactions defined as investing or financing activities, such as amounts received to settle lawsuits, proceeds of certain insurance settlements, and cash refunds from suppliers.
Cash outflows for operating activities affect items that appear on the income statement and include payments: (1) to acquire inventory; (2) to other suppliers and employees for other goods or services; (3) to lenders and other creditors for interest; (4) for purchases of trading securities; and (5) all other cash payments that do not arise from transactions defined as investing or financing activities, such as taxes and payments to settle lawsuits, cash contributions to charities, and cash refunds to customers.
Investing activities generally include transactions involving the acquisition or disposal of noncurrent assets. Thus, cash inflows from investing activities include cash received from: (1) the sale of property, plant, and equipment; (2) the sale of available-for-sale and held-to-maturity securities; and (3) the collection of long-term loans made to others. Cash outflows for investing activities include cash paid: (1) to purchase property, plant, and equipment; (2) to purchase available-for-sale and held-to-maturity securities; and (3) to make long-term loans to others.
Financing activities generally include the cash effects (inflows and outflows) of transactions and other events involving creditors and owners. Cash inflows from financing activities include cash received from issuing capital stock and bonds, mortgages, and notes, and from other short- or long-term borrowing. Cash outflows for financing activities include payments of cash dividends or other distributions to owners (including cash paid to purchase treasury stock) and repayments of amounts borrowed. Payment of interest is not included because interest expense appears on the income statement and is, therefore, included in operating activities. Cash payments to settle accounts payable, wages payable, and income taxes payable are not financing activities. These payments are included in the operating activities section.
Information about all material investing and financing activities of an enterprise that do not result in cash receipts or disbursements during the period appear in a separate schedule, rather than in the statement of cash flows. The disclosure may be in narrative form. For instance, assume a company issued a mortgage note to acquire land and buildings.
Financial Statements: Interconnectivity
Watch the following video, and pay special attention to the interconnection between the four financial statements required by GAAP.


11.2: Systems of Linear Equations with Three Variables
- Solve systems of three equations in three variables.
- Identify inconsistent systems of equations containing three variables.
- Express the solution of a system of dependent equations containing three variables.
John received an inheritance of \($12,000\) that he divided into three parts and invested in three ways: in a money-market fund paying \(3\%\) annual interest; in municipal bonds paying \(4\%\) annual interest; and in mutual funds paying \(7\%\) annual interest. John invested \($4,000\) more in municipal funds than in municipal bonds. He earned \($670\) in interest the first year. How much did John invest in each type of fund?
Figure \(\PageIndex{1}\): (credit: “Elembis,” Wikimedia Commons)
Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visual gymnastics.
Solving Systems of Three Equations in Three Variables
In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution \((x,y,z)\), which we call an ordered triple. A system in upper triangular form looks like the following:
\[\begin{align*} Ax+By+Cz &= D \nonumber \\[4pt] Ey+Fz &= G \nonumber \\[4pt] Hz &= K \nonumber \end{align*} \nonumber\]
The third equation can be solved for \(z\),and then we back-substitute to find \(y\) and \(x\). To write the system in upper triangular form, we can perform the following operations:
- Interchange the order of any two equations.
- Multiply both sides of an equation by a nonzero constant.
- Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple \({(x,y,z)}\). Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.
Figure \(\PageIndex{2}\) and Figure \(\PageIndex{3}\) illustrate possible solution scenarios for three-by-three systems.
- Systems that have a single solution are those which, after elimination, result in a solution set consisting of an ordered triple \({(x,y,z)}\). Graphically, the ordered triple defines a point that is the intersection of three planes in space.
- Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as \(0=0\). Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space.
- Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as \(3=0\). Graphically, a system with no solution is represented by three planes with no point in common.
Figure \(\PageIndex{2}\): (a)Three planes intersect at a single point, representing a three-by-three system with a single solution. (b) Three planes intersect in a line, representing a three-by-three system with infinite solutions.
Figure \(\PageIndex{3}\): All three figures represent three-by-three systems with no solution. (a) The three planes intersect with each other, but not at a common point. (b) Two of the planes are parallel and intersect with the third plane, but not with each other. (c) All three planes are parallel, so there is no point of intersection.
Determine whether the ordered triple \((3,−2,1)\) is a solution to the system.
\[\begin{align*} x+y+z &= 2 \nonumber \\[4pt] 6x−4y+5z &= 31 \nonumber \\[4pt] 5x+2y+2z &= 13 \nonumber \end{align*} \nonumber\]
Solution
We will check each equation by substituting in the values of the ordered triple for \( x,y\), and \(z\).
\[\begin{array}{rrr} { \text{} \nonumber \\[4pt] x+y+z=2 \nonumber \\[4pt] (3)+(−2)+(1)=2 \nonumber \\[4pt] \text{True}} & {6x−4y+5z=31 \nonumber \\[4pt] 6(3)−4(−2)+5(1)=31 \nonumber \\[4pt] 18+8+5=31 \nonumber \\[4pt] \text{True} } & { 5x+2y+2z = 13 \nonumber \\[4pt] 5(3)+2(−2)+2(1)=13 \nonumber \\[4pt] 15−4+2=13 \nonumber \\[4pt] \text{True}} \end{array}\]
The ordered triple \((3,−2,1)\) is indeed a solution to the system.
- Pick any pair of equations and solve for one variable.
- Pick another pair of equations and solve for the same variable.
- You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.
- Back-substitute known variables into any one of the original equations and solve for the missing variable.
Find a solution to the following system:
\[\begin{align} x−2y+3z=9 \; &(1) \nonumber \\[4pt] −x+3y−z=−6 \; &(2) \nonumber \\[4pt] 2x−5y+5z=17 \; &(3) \nonumber \end{align} \nonumber\]
Solution
There will always be several choices as to where to begin, but the most obvious first step here is to eliminate \(x\) by adding equations (1) and (2).
\[\begin{align} x−2y+3z=9 \; \; &(1) \nonumber \\[4pt] \underline{−x+3y−z=−6 }\; \; &(2) \nonumber \\[4pt] y+2z=3 \;\; &(4) \nonumber \end{align} \nonumber\]
The second step is multiplying equation (1) by \(−2\) and adding the result to equation (3). These two steps will eliminate the variable \(x\).
\[\begin{align} −2x+4y−6z=−18\; &(1) \;\;\;\; \text{ multiplied by }−2 \nonumber \\[4pt] \underline{2x−5y+5z=17} \; & (3) \nonumber \\[4pt]−y−z=−1 \; &(5) \nonumber \end{align} \nonumber\]
In equations (4) and (5), we have created a new two-by-two system. We can solve for \(z\) by adding the two equations.
\[\begin{align} y+2z=3 \; &(4) \nonumber \\[4pt] \underline{−y−z=−1} \; & (5) \nonumber \\[4pt] z=2 \; & (6) \nonumber \end{align} \nonumber\]
Choosing one equation from each new system, we obtain the upper triangular form:
\[\begin{align} x−2y+3z=9 \; &(1) \nonumber \\[4pt] y+2z =3 \; &(4) \nonumber \\[4pt] z=2 \; &(6) \nonumber \end{align} \nonumber\]
Next, we back-substitute \(z=2\) into equation (4) and solve for \(y\).
\[\begin{align} y+2(2) &=3 \nonumber \\[4pt] y+4 &= 3 \nonumber \\[4pt] y &= −1 \nonumber \end{align} \nonumber\]
Finally, we can back-substitute \(z=2\) and \(y=−1\) into equation (1). This will yield the solution for \(x\).
\[\begin{align} x−2(−1)+3(2) &= 9 \nonumber \\[4pt] x+2+6 &=9 \nonumber \\[4pt] x &= 1 \nonumber \end{align} \nonumber\]
The solution is the ordered triple \((1,−1,2)\). See Figure \(\PageIndex{4}\).
Figure \(\PageIndex{4}\)
In the problem posed at the beginning of the section, John invested his inheritance of \($12,000\) in three different funds: part in a money-market fund paying \(3\%\) interest annually; part in municipal bonds paying \(4\%\) annually; and the rest in mutual funds paying \(7\%\) annually. John invested \($4,000\) more in mutual funds than he invested in municipal bonds. The total interest earned in one year was \($670\). How much did he invest in each type of fund?
Solution
To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investment amounts:
\[\begin{align} x &= \text{amount invested in money-market fund} \nonumber \\[4pt] y &= \text{amount invested in municipal bonds} \nonumber \\[4pt] z &= \text{amount invested in mutual funds} \nonumber \end{align} \nonumber\]
The first equation indicates that the sum of the three principal amounts is \($12,000\).
\[x+y+z=12,000 \nonumber\]
We form the second equation according to the information that John invested \($4,000\) more in mutual funds than he invested in municipal bonds.
\[z=y+4,000 \nonumber\]
The third equation shows that the total amount of interest earned from each fund equals \($670\).
\[0.03x+0.04y+0.07z=670 \nonumber\]
Then, we write the three equations as a system.
\[\begin{align} x+y+z &=12,000 \nonumber \\[4pt] −y+z &= 4,000 \nonumber \\[4pt] 0.03x+0.04y+0.07z &= 670 \nonumber \end{align} \nonumber\]
To make the calculations simpler, we can multiply the third equation by \(100\). Thus,
\[\begin{align} x+y+z &=12,000 \; &(1) \nonumber \\[4pt] −y+z &= 4,000 \; &(2) \nonumber \\[4pt] 3x+4y+7z &= 67,000 \; &(3) \nonumber \end{align} \nonumber\]
Step 1. Interchange equation (2) and equation (3) so that the two equations with three variables will line up.
\[\begin{align} x+y+z &= 12,000 \nonumber \\[4pt] 3x+4y+7z &= 67,000 \nonumber \\[4pt] −y+z &= 4,000 \nonumber \end{align} \nonumber\]
Step 2. Multiply equation (1) by \(−3\) and add to equation (2). Write the result as row 2.
\[\begin{align} x+y+z &= 12,000 \nonumber \\[4pt] y+4z &= 31,000 \nonumber \\[4pt] −y+z &= 4,000 \nonumber \end{align} \nonumber\]
Step 3. Add equation (2) to equation (3) and write the result as equation (3).
\[\begin{align} x+y+z &= 12,000 \nonumber \\[4pt] y+4z &= 31,000 \nonumber \\[4pt] 5z &= 35,000 \nonumber \end{align} \nonumber\]
Step 4. Solve for \(z\) in equation (3). Back-substitute that value in equation (2) and solve for \(y\). Then, back-substitute the values for \(z\) and \(y\) into equation (1) and solve for \(x\).
\[\begin{align} 5z &= 35,000 \nonumber \\[4pt] z &= 7,000 \nonumber \\[4pt] \nonumber \\[4pt] y+4(7,000) &= 31,000 \nonumber \\[4pt] y &=3,000 \nonumber \\[4pt] \nonumber \\[4pt] x+3,000+7,000 &= 12,000 \nonumber \\[4pt] x &= 2,000 \nonumber \end{align} \nonumber\]
John invested \($2,000\) in a money-market fund, \($3,000\) in municipal bonds, and \($7,000\) in mutual funds.
Solve the system of equations in three variables.
\[\begin{align} 2x+y−2z &= −1 \nonumber \\[4pt] 3x−3y−z &= 5 \nonumber \\[4pt] x−2y+3z &= 6 \nonumber \end{align} \nonumber\]
- Answer
-
\((1,−1,1)\)
Identifying Inconsistent Systems of Equations Containing Three Variables
Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as \(3=7\) or some other contradiction.
Solve the following system.
\[\begin{align} x−3y+z &=4 \label{4.1}\\[4pt] −x+2y−5z &=3 \label{4.2} \\[4pt] 5x−13y+13z &=8 \label{4.3} \end{align} \nonumber\]
Solution
Looking at the coefficients of \(x\), we can see that we can eliminate \(x\) by adding Equation \ref{4.1} to Equation \ref{4.2}.
\[\begin{align} x−3y+z = 4 &(1) \nonumber \\[4pt] \underline{−x+2y−5z=3} & (2) \nonumber \\[4pt] −y−4z =7 & (4) \nonumber \end{align} \nonumber\]
Next, we multiply equation (1) by \(−5\) and add it to equation (3).
\[\begin{align} −5x+15y−5z =−20 & (1) \;\;\;\;\; \text{multiplied by }−5 \nonumber \\[4pt] \underline{5x−13y+13z=8} &(3) \nonumber \\[4pt] 2y+8z=−12 &(5) \nonumber \end{align} \nonumber\]
Then, we multiply equation (4) by 2 and add it to equation (5).
\[\begin{align} −2y−8z=14 & (4) \;\;\;\;\; \text{multiplied by }2 \nonumber \\[4pt] \underline{2y+8z=−12} & (5) \nonumber \\[4pt] 0=2 & \nonumber \end{align} \nonumber\]
The final equation \(0=2\) is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.
Analysis
In this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.
Solve the system of three equations in three variables.
\[\begin{align} x+y+z &= 2 \nonumber \\[4pt] y−3z &=1 \nonumber \\[4pt] 2x+y+5z &=0 \nonumber \end{align} \nonumber\]
- Answer
-
No solution.
Expressing the Solution of a System of Dependent Equations Containing Three Variables
We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.
Find the solution to the given system of three equations in three variables.
\[\begin{align} 2x+y−3z &= 0 &(1) \nonumber \\[4pt] 4x+2y−6z &=0 &(2) \nonumber \\[4pt] x−y+z &= 0 &(3) \nonumber \end{align} \nonumber\]
Solution
First, we can multiply equation (1) by \(−2\) and add it to equation (2).
\[\begin{align} −4x−2y+6z =0 & (1) \;\;\;\;\; \text{multiplied by }−2 \nonumber \\[4pt] \underline{4x+2y−6z=0} & (2) \nonumber \\[4pt] 0=0& \nonumber \end{align} \nonumber\]
We do not need to proceed any further. The result we get is an identity, \(0=0\), which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by \(−2\), and adding it to equation (1). We then perform the same steps as above and find the same result, \(0=0\).
When a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have
\[\begin{align} 2x+y−3z &= 0 \nonumber \\[4pt]x−y+z &= 0 \nonumber \\[4pt] 3x−2z &= 0 \nonumber \end{align} \nonumber\]
We then solve the resulting equation for \(z\).
\[\begin{align} 3x−2z &= 0 \nonumber \\[4pt] z &= \dfrac{3}{2}x \nonumber \end{align} \nonumber\]
We back-substitute the expression for \(z\) into one of the equations and solve for \(y\).
\[\begin{align} 2x+y−3 (\dfrac{3}{2}x) &= 0 \nonumber \\[4pt] 2x+y−\dfrac{9}{2}x &= 0 \nonumber \\[4pt] y &= \dfrac{9}{2}x−2x \nonumber \\[4pt] y &=\dfrac{5}{2}x \nonumber \end{align} \nonumber\]
So the general solution is \(\left(x,\dfrac{5}{2}x,\dfrac{3}{2}x\right)\). In this solution, \(x\) can be any real number. The values of \(y\) and \(z\) are dependent on the value selected for \(x\).
Analysis
As shown in Figure \(\PageIndex{5}\), two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.
Figure \(\PageIndex{5}\)
No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of \(x\) and if needed \(x\) and \(y\).
Solve the following system.
\[\begin{align} x+y+z &= 7 \nonumber \\[4pt] 3x−2y−z &= 4 \nonumber \\[4pt] x+6y+5z &= 24 \nonumber \end{align} \nonumber\]
- Answer
-
Infinite number of solutions of the form \((x,4x−11,−5x+18)\).
Access these online resources for additional instruction and practice with systems of equations in three variables.
Key Concepts
- A solution set is an ordered triple {(x,y,z)} that represents the intersection of three planes in space. See Example \(\PageIndex{1}\).
- A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See Example \(\PageIndex{2}\).
- Systems of three equations in three variables are useful for solving many different types of real-world problems. See Example \(\PageIndex{3}\).
- A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See Example \(\PageIndex{4}\).
- Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
- A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See Example \(\PageIndex{5}\).
- Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.
Contributors
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Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


8.1: Introduction to Cancer and Phases of Cancer Therapy
By the end of this section, you should be able to:
- 8.1.1 Describe cancer and cancer development.
- 8.1.2 Discuss contributing theories of environmental versus genetic etiologies of cancer development.
- 8.1.3 Identify the characteristics of cancer cells.
- 8.1.4 Describe different types of cancer.
Cancer Development
Each day, the human body experiences cell mutations that result in a change in cellular structure, which may lead to cancer development. When this happens, the intracellular machinery may be able to repair the mutation, or the body’s immune system recognizes these cells as abnormal and attacks them. These are two innate defenses against cancer development. When either defense fails, cancer development can occur. There are multiple theories about the causes of cancer development. The most common theories are based on environmental exposures and genetic predispositions as causes of cancer development.
Environmental and Genetic Factors
Environmental exposures are outside factors that can cause cell mutations when the body is exposed to them. Most commonly, these factors include exposure to tobacco products, benzenes, petroleum-based products, asbestos, and certain drugs, including chemotherapies. There are many other substances that are classified as carcinogens, or cancer-causing agents. Somatic mutations occur from carcinogen exposures after birth. They do not develop from genetic mutations and do not affect germ cells that become ova and sperm.
Link to Learning
One of the earliest findings of cancer and other diseases directly linked to environmental exposures occurred in the small community of Love Canal, Niagara Falls, in upstate New York. The story of Love Canal, while very unfortunate, brought about widespread change in environmental protection and regulation of toxic substances.
Genetic factors are those internal predispositions to cellular changes that result in cancer development. For example, clients who inherit the breast cancer oncogenes ( BRCA1 and BRCA2 ) have a much higher chance of developing breast cancer than clients without these mutations (Centers for Disease Control and Prevention [CDC], 2020). These are considered to be germline mutations that develop in the eggs or sperm of a parent and are passed on to offspring.
Link to Learning
Oncogenes and Breast Cancer
Because genetic mutations that cause specific cancers to develop are present at birth, screening for the presence of some of these oncogenes is now possible. When these oncogenes are identified before cancer has developed, clients now have options for improved surveillance and early detection and intervention. This promotes better outcomes for those clients who were born with hereditary gene mutations. The breast cancer antigen mutations 1 and 2 (BRCA1, BRCA2) are two of the most well-understood oncogenes.
Additional theories are based on immune system failure and the effects that physical and mental stress have on the body’s ability to defend itself. Cell mutations occur frequently throughout the human body, but these cells are usually targeted and either repaired or destroyed by the immune system. When the immune system fails to recognize mutated cells, these cells continue replication, forming a tumor. Other illnesses, physical stress, and mental distress are some factors that decrease the effectiveness of the immune-system functions. When considering theories of cancer development, most researchers agree that a combination of causative factors, rather than one single theory, contributes to cancer development (Mbemi et al., 2020).
Cellular Changes
Once a cancer-causing mutation occurs, if it is unable to be repaired, the mutated cells continue to divide, resulting in the development of a tumor or malignancy (see Figure 8.2). These cells lose many of the regulatory characteristics of normal cells, including contact inhibition and a regulated rate of mitosis. Tumors also have the property of neoangiogenesis , which is the ability to grow new blood vessels to support the metabolic needs associated with the abnormal growth. As these cells continue to grow, they can result in physical changes that greatly affect the function of the body.
Pain, compression, nutritional deficiencies, weight loss, and fluid and electrolyte imbalances are some of the effects that can occur with cancer development. However, tumors can also go unnoticed for many years. As growth continues, cancerous cells can metastasize to distant sites through direct extension into surrounding tissues, through seeding, and by embolization into the lymph and circulatory systems.
Cancer Types
Cancers are classified either by the type of tissue in which the cancer originates (histological type) or by the location in the body where the cancer originated (primary site) (National Cancer Institute, n.d.). Health care providers tend to refer to a cancer’s histological type, while clients more frequently refer to the cancer’s location. Many types of cancer will form tumors. Solid tumors are cancers that form in tissues of bones, skin, organs, and muscles. Liquid, or hematologic, tumors arise in the bone marrow and involve blood and lymphatic cells. Typically, regardless of tumor type, pharmacologic treatment will involve medications with systemic effects.
Solid Tumors
Solid tumors develop as a mass of cancer cells that originate in a specific area of the body. Adenocarcinomas and squamous cell carcinomas are two common pathological types of solid tumors. Adenocarcinomas develop in glandular and epithelial tissues, while squamous cell carcinomas arise in the skin or in the lining of the respiratory and gastrointestinal system (National Cancer Institute, n.d.). Another type of solid tumor that is less common is a sarcoma. These tumors may form in soft tissues including fat, muscles, nerves, vessels, and skin. Sarcomas may also form in bones and most often affect young adults. In the United States, lung, prostate, and breast cancers are the leading causes of cancer-related mortality (National Cancer Institute, n.d.).
 These cancers are often called liquid tumors because they originate in the bone marrow and flow in the bloodstream. In the United States, non-Hodgkin lymphoma , leukemia , and multiple myeloma are the three most common hematologic cancers, respectively. These types of tumors present multiple issues because they exist in the vascular system, circulating through all parts of the body. When these cells are destroyed by chemotherapy, cellular debris and electrolytes are released into the blood. This may result in tumor lysis syndrome (TLS), a life-threatening condition that is characterized by acidosis, hyperkalemia, hyperphosphatemia, and hypocalcemia. When chemotherapy is given for a blood tumor in which high numbers of cancer cells are in the blood, clients must be pretreated with hydration therapy, allopurinol or rasburicase , and management of electrolytes. While tumor lysis occurs most commonly in hematologic cancers, it may occasionally occur with solid tumors that are very large and bulky.
Link to Learning
Tumor Lysis Syndrome
Tumor lysis syndrome is considered an oncologic or medical emergency, which without immediate recognition and treatment can result in fatal outcomes. The Cleveland Clinic has provided important updates about this syndrome that can occur within hours of cancer treatment.


14.2: Toussaint’s Constitution (Excerpt)
CONSTITUTION OF 1801
The representatives of the colony of
Saint-Domingue
, gathered in Central Assembly, have arrested and established the constitutional bases of the regime of the French colony of Saint-Domingue as follows:
TITLE I
Of the Territory
Art. 1. – Saint-Domingue in its entire expanse, and Samana, La Tortue , La Gonave, Les Cayemites, L’Ile-a-Vache, La Saone and other adjacent islands form the territory of a single colony, which is part of the French Empire, but ruled under particular laws.
Art. 2. – The territory of this colony is divided in departments, arrondissements (districts) and parishes.
TITLE II
Of the Inhabitants
Art. 3. – There cannot exist slaves on this territory, servitude is therein forever abolished.
Art. 4. – All men, regardless of color, are eligible to all employment.
Art. 5. – There shall exist no distinction other than those based on virtue and talent, and other superiority afforded by law in the exercise of a public function.
The law is the same for all whether in punishment or in protection.
TITLE III
Of the Religion
Art. 6. – The catholic, apostolic, roman faith shall be the only publicly professed faith.
Art. 7. – Each parish shall provide to the maintaining of religious cult and of its ministers. The wealth of the factories shall be especially allocated to this expense, and the presbyteries to the housing of ministers.
Art. 8. – The governor of the colony shall assign to each minister of the religion the extent of his spiritual administration, and said ministers can never, under any circumstance, form a corps in the colony.
TITLE IV
Of the Mores
Art. 9. – Marriage, by its civic and religious institution, tend to the purity of mores; spouses who will practice the virtues required by their condition shall always be distinguished and especially protected by the government.
Art. 10. – Divorce shall not take place in the colony.
Art. 11. – Laws that will tend to expand and maintain social virtues, and to encourage and cement family bonding shall fix condition and rights of children born in wedlock.
TITLE V
Of Men in Society
Art. 12. – The Constitution guarantees freedom and individual security. No one shall be arrested unless a formally expressed mandate, issued from a functionary to whom the law grants the right to order arrest and detention in a publicly designated location.
Art. 13. – Property is sacred and inviolable. All people, either by himself, or by his representatives, has the free right to dispose and to administer property that is recognized as belonging to him. Anyone who attempts to deny this right shall become guilty of crime towards society and responsible towards the person troubled in his property.
TITLE VI
Of Cultures and Commerce
Art. 14. – The colony being essentially agricultural cannot suffer the least disruption in the works of its cultivation.
Art. 15. – Each habitation shall constitute a manufacture that requires the gathering of cultivators and workers; it shall represent the quiet haven of an active and constant family, of which the owner of the land or his representative shall be the father.
Art. 16. – Each cultivator and each worker is a member of the family and shares in parts of the revenues.
Every change in domicile on the part of the cultivator carries the ruin of the cultivation. In order to repress a vice as disruptive to the colony as it is to public order, the governor issues all policy requirements necessary in the circumstances and in conformance with the bases of rules of police of 20 Vendémiaire , year IX [1801], and of the proclamation of the following l9th Pluviôse of the Chief General Toussaint-Louverture .
Art. 17. – The introduction of cultivators indispensable to the reestablishment and to the growth of agriculture shall take place in Saint-Domingue . The Constitution charges the Governor to take convenient measures to encourage and favor the increase in manpower, to stipulate and balance the diverse interests, to ensure and guarantee the execution of respective engagements resulting from this introduction.
Art. 18. – Commerce in the colony consists uniquely of exchange goods produced on its territory; consequently, the introduction of goods similar in nature is and shall remains prohibited.
TITLE VIII
Of the Government
Art. 27. – The administrative direction of the government shall be entrusted to a Governor who corresponds directly with the government of the Metropole, on all matters relative to the interests of the colony.
Art. 28. – The Constitution nominates the citizen Toussaint-Louverture , Chief General of the army of Saint-Domingue , and, in consideration for important services rendered to the colony, in the most critical circumstances of the revolution, and upon the wishes of the grateful inhabitants, he is entrusted the direction thereof for the remainder of his glorious life.
Art. 29. – In the future, each governor shall be nominated for five years, and shall continue every five years for reasons of his good administration.
Art. 30. – In order to strengthen the tranquility that the colony owes to steadfastness, activity, indefatigable zeal and rare virtues of the General Toussaint-Louverture , and in sign of the unlimited trust of the inhabitants of Saint-Domingue , the Constitution attribute exclusively to this general the right to designate the citizen who, in the unfortunate event of the general’s death, shall immediately replace him. This choice shall remain secret; it shall be cosigned under sealed envelope and to be opened only by the Central Assembly, in presence of all active generals and chief commanders of departments of the army of Saint-Domingue .
The Governor Toussaint-Louverture shall take all necessary precautionary measures to let the Central Assembly know the depository of this important envelope.
Art. 31. – The citizen that shall be chosen by the Governor Toussaint-Louverture to take the direction of the government upon his death, shall swear in front of the Central Assembly to execute the Constitution of Saint-Domingue and to remain attached to the French government, and shall be immediately installed in his functions; all shall be in presence of active generals and chief commanders of departments of the army of Saint-Domingue , who all, individually and without delay, shall swear obedience to the orders of the new Governors Saint-Domingue .
Art. 32. – At least one month before the expiration of the five years fixed for the administration of each General, the one in central function, jointly with the active-duty Generals and Chief Commanders of Departments, shall meet at the ordinary place of hearing of the Central Assembly. to the effect of nominating, concurrently with the members of this Assembly, the new Governor or continue the administration of the one who is in function.
Art. 33. – Failure on the part of a Governor in function to convoke constitutes a manifest infraction to the Constitution. In such circumstance, the highest ranked General or the senior General of equal rank, who is in active service in the colony, shall take, of right, if provisionally, the control the govermnent.
This General shall convoke immediately the other General in active duty, the Chief Commanders of Departments and the members of the Central Assembly, who shall all obey the convocation, to the effect of proceeding concurrently to the nomination of a new Governor.
In the event of death, resignation or other vacancy by a Governor before the expiration of his mandate, the Government passes as well provisionally to the highest ranked General, or the senior General of equal rank who shall convoke, to the same ends as above, the members of the Central Assembly, the active-duty Generals and Chief Commanders of Departments.
Art. 34.– The Governor shall seal and promulgate the laws; he nominates to all civilian and military employment. He is the chief commander of the armed forces and is charged with its organization; State vessels in station at the shores of the colony receive orders from him.
He shall determine the divisions of the territory in manners most conform to internal relations. He watches and provides, according to the law, for internal and external security of the colony, and given that the state of war is a state of abandonment, malaise and nullity for the colony, the Governor is charged to take in those circumstances measures he deems necessary to ensure the subsistence and the supply of goods of all sorts to the colony.
Art. 35. – He shall exercise the general police of inhabitants and of the factories, and enforces the obligations of owners, farmers and of their representatives towards cultivators and workers, and the duty of cultivators towards owners, farmers or their representatives.
Art. 36. – He shall propose laws to the Central Assembly, as well as changes to the Constitution that experience may necessitate.
Art. 37. – He shall direct, supervise the collection, the payments and the use of finances of the colony, and shall give, to this effect, any and all orders.
Art. 38. – He shall present, every two years, to the Central Assembly the conditions of receipts and disbursements of each department, year by year.
Art. 39. – He shall supervise and censor by the authority of his commissaries, all writings designed for printing on the island he shall cause to be suppressed all those coming from abroad that would tend to corrupt mores or trouble the new colony; he shall punish the authors or colporteurs, according to the severity of the situation.
Art. 40. – If the Governor is informed of some plot against the tranquility of the colony, he shall immediately proceed to the arrest of the presumed authors, instigators or accomplices; after having them undergo extra-judiciary questioning, he shall cite them in front of a competent tribunal.
Art. 41. – The salary of the Governor is fixed at the present time at 300.000 Francs. His honor guard shall be charged to the colony.
TITLE IX
Of the Courts
Art. 42. – Citizen shall have an inalienable right to be judged by arbiters at their choice.
Art. 43. – No authority shall suspend nor impeach the execution ofjudgments rendered by the Courts.
Art. 44. – Justice shall be administered in the colony by Courts of first instance and by Courts of appeal. The law determines their organization, their number, their competence and the territory of each Courtís jurisdiction.
These tribunals, according to their degree of jurisdiction, shall recognize all civil and criminal affairs.
Art. 45. – There shall exist for the colony a Court of Cassation that shall pronounce on demands of annulments against judgments rendered by Appeal Courts, and issue opinions against an entire tribunal. This court does not hear the facts of the cases, but overturn judgments rendered on procedures in which the forms have been violated; or that contain some express contravention [infringement] to the law, and shall return the facts of the process to the tribunal in question.
Art. 46. – Judges of divers Courts conserve their function for life, unless they are condemned for forfeiture. Commissaries of the government can be revoked.
Art. 47. – Military misdemeanors shall be submitted to special tribunals and subject to special judgments.
These special Courts also hear cases of theft, abduction, domicile violation, murder, assassination, arson, rape, plotting and mutiny.
The organization of these Courts pertains to the Governor of the colony.
TITLE XI
Of the Armed Forces
Art. 52. – The Armed Forces are essentially obedient, they can never deliberate; they are at the disposition of the Governor who can mobilize them only to maintain public order, protection due to all citizens, and the defense of the colony.
Art. 53. – They are divided in paid colonial guard and unpaid colonial guard.
Art. 54. – The unpaid colonial guard shall not go out of the limits of its parish unless there is a case of imminent danger, and upon the order and the responsibility of the local military commander.
Outside of its parish it shall be compensated; and shall be submitted, in this case, to the military discipline, and in all other case, is only subject to the law.
Art. 55. – The state police force of the colony shall be part of the Armed Forces; it shall be divided in a mounted force and a pedestrian force. The mounted force is instituted for the high police of security of the countryside; it has the charge of the wealth of the colony.
The pedestrian force is instituted for the police of cities and towns; it shall be at the charge of the city or town for which it performs services.
Art. 56. – The army is recruited upon the proposition the Governor makes to the Central Assembly, according to the mode established by law.
TITLE XIII
General Dispositions
Art. 63. – The residence of any person shall constitute an inviolable abode. During nighttime, no one shall have the right to enter therein unless in case of fire, flooding or upon request from within. During the day, one shall have access for a special determined object or, by a law, or by order issued from a public authority.
Art. 64. – For a lawful arrest to be executed; it must
- 1) formally express the motive of the arrest and the law in virtue of which it is ordered;
- 2) be issued from a functionary whom the law formally empowers to do so;
- 3) presented to the person in form of copy of the warrant.
Art. 65. – Anyone who, without authority of the law to make an arrest, gives, signs, executes or causes to be executed the arrest of a person, shall be guilty of the crime of arbitrary detention.
Art. 66. – Any person shall have the right to address individual petitions to all constitutional authority and especially to the Governor.
Art. 67. – There cannot exist in the colony corporations or associations that are contrary to public order.
No citizen association shall be qualified as popular society. All seditious gathering shall be dissipated immediately, first by way of verbal order and, if necessary, by development of armed force.
Art. 68. – Any person shall have the faculty to form particular establishments of education and instruction for the youth under the authorization and the supervision of municipal administrations.
Art. 69. – The law supervises especially all professions dealing with public mores, public safety, health and fortune of citizens.
Art. 70. – The law provides for awards to inventors of rural machines, or for the preservation of the exclusive ownership of their discoveries.
Art. 71. – There shall exist in the colony uniformity of weights and measures.
Art. 72. – It shall be given, by the Governor, in the name of the colony, awards to warriors who will have rendered exceptional services while fighting for the common defense.
‘Art. 73. – Absentee owners, for whatever reason, conserve all their rights to properties belonging to them and situated in the colony; it suffices, to remove any sequestration that might have been imposed, to reintroduce their titles of ownership and; in default of title thereof, supplementary acts whose formula is determined by law. Exempt of this disposition are, nevertheless, those who might been inscribed and maintained on the general list of emigrants of France; their properties shall continue, in this case, to be administered as colonial domains until their removal from the list.
Art. 74. – The colony proclaims, as guarantee of public law, that all leases [beaux? /as spelled in original] of legally leased properties by the administration shall have their full effect, if the contracting parties prefer not to compromise with owners or their representatives who would obtain the return of their sequestered goods.
Art. 75.– It proclaims that it is on the respect of the citizen [personne] and of the properties that rest agriculture, all productions, all means of employment and all social order.
Art. 76. – It proclaims that any citizen owes services to the land that nourishes him or that guarantees his rights, and in regard to those [services] that shall have been collected, at a later time, they shall be exactable and reimbursed in the year that follows the lifting of sequestration of goods.
Art. 77. – The Chief General Toussaint-Louverture is and shall remain charged with sending the present Constitution to be sanctioned by the French government; nevertheless, and given the absence of laws, the urgency to exit from this condition of peril, the necessity to promptly reestablish agriculture and the unanimous wishes pronounced by the inhabitants of Saint-Domingue, the Chief General is and remains invited, in the name of public good, to proceed with its execution in all areas of the territory of the colony.
Made at Port-Republican , this 19th Floréal year IX of the French Republic, one and indivisible.
- Signed: Borgella, President
- Raymond Collet Gaston Nogérée
- Lacour,
- Roxas,
- Munos,
- Mancebo,
- E. Viert, secretary
After having taken knowledge of the Constitution, I give it my approval. The invitation of the Central Assembly is for me an order; consequently, I shall pass it to the French government in order to obtain its sanction; as for its execution in the colony, the wish expressed by the Central Assembly shall be fulfilled as well and executed.
Given at Cap Français , this 14 Messidor , year IX [July 3, 1801] of the French Republic, one and indivisible.
The Chief General:
Signed: Toussaint-Louverture


2.3.3: Solving Problems about Proportional Relationships
Lesson
Let's solve problems about proportional relationships.
Exercise \(\PageIndex{1}\): What Do You Want to Know?
Consider the problem: A person is running a distance race at a constant rate. What time will they finish the race?
What information would you need to be able to solve the problem?
Exercise \(\PageIndex{2}\): Info Gap: Biking and Rain
Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.
If your teacher gives you the problem card :
- Silently read your card and think about what information you need to be able to answer the question.
- Ask your partner for the specific information that you need.
-
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently.
- Read the data card and discuss your reasoning.
If your teacher gives you the data card :
- Silently read your card.
-
Ask your partner
“What specific information do you need?”
and wait for them to
ask
for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
- Read the problem card and solve the problem independently.
- Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Exercise \(\PageIndex{3}\): Moderating Comments
A company is hiring people to read through all the comments posted on their website to make sure they are appropriate. Four people applied for the job and were given one day to show how quickly they could check comments.
- Person 1 worked for 210 minutes and checked a total of 50,000 comments.
- Person 2 worked for 200 minutes and checked 1,325 comments every 5 minutes.
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Person 3 worked for 120 minutes, at a rate represented by \(c=331t\),
where is the number of comments checked and is the time in minutes. - Person 4 worked for 150 minutes, at a rate represented by \(t=\left(\frac{3}{800}\right)c\).
- Order the people from greatest to least in terms of total number of comments checked.
- Order the people from greatest to least in terms of how fast they checked the comments.
Are you ready for more?
- Write equations for each job applicant that allow you to easily decide who is working the fastest.
- Make a table that allows you to easily compare how many comments the four job applicants can check.
Summary
Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between quantities of interest.
- When a bird is flying at a constant speed, then there is a proportional relationship between the flying time and distance flown.
- If water is filling a tub at a constant rate, then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up.
- If an aardvark is eating termites at a constant rate, then there is proportional relationship between the number of termites the aardvark has eaten and the time since it started eating.
Sometimes we are presented with a situation, and it is not so clear whether a proportional relationship is a good model. How can we decide if a proportional relationship is a good representation of a particular situation?
- If you aren’t sure where to start, look at the quotients of corresponding values. If they are not always the same, then the relationship is definitely not a proportional relationship.
- If you can see that there is a single value that we always multiply one quantity by to get the other quantity, it is definitely a proportional relationship.
After establishing that it is a proportional relationship, setting up an equation is often the most efficient way to solve problems related to the situation.
Practice
Exercise \(\PageIndex{4}\)
For each situation, explain whether you think the relationship is proportional or not. Explain your reasoning.
- The weight of a stack of standard 8.5x11 copier paper vs. number of sheets of paper.
- The weight of a stack of different-sized books vs. the number of books in the stack.
Exercise \(\PageIndex{5}\)
Every package of a certain toy also includes 2 batteries.
- Are the number of toys and number of batteries in a proportional relationship? If so, what are the two constants of proportionality? If not, explain your reasoning.
- Use \(t\) for the number of toys and \(b\) for the number of batteries to write two equations relating the two variables.
\(b=\qquad\qquad t=\)
Exercise \(\PageIndex{6}\)
Lin and her brother were born on the same date in different years. Lin was 5 years old when her brother was 2.
-
Find their ages in different years by filling in the table.
Lin's age Her brother's age \(5\) \(2\) \(6\) \(15\) \(25\) Table \(\PageIndex{1}\) - Is there a proportional relationship between Lin's age and her brother's age? Explain your reasoning.
Exercise \(\PageIndex{7}\)
A student argues that \(y=\frac{x}{9}\) does not represent a proportional relationship between \(x\) and \(y\) because we need to multiply one variable by the same constant to get the other one and not divide it by a constant. Do you agree or disagree with this student?
(From Unit 2.3.2)
Exercise \(\PageIndex{8}\)
Quadrilateral A has side lengths 3, 4, 5, and 6. Quadrilateral B is a scaled copy of Quadrilateral A with a scale factor of 2. Select all of the following that are side lengths of Quadrilateral B.
- \(5\)
- \(6\)
- \(7\)
- \(8\)
- \(9\)
(From Unit 1.1.3)


3.1: Finding the Boltzmann Equation
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The probabilities of the energy levels of a constant-temperature system at equilibrium must depend only on the intensive variables that serve to characterize the equilibrium state. In Section 20.8 , we introduce the principle of equal a priori probabilities, which asserts that any two microstates of an isolated system have the same probability. From the central limit theorem, we infer that an isolated system is functionally equivalent to a constant-temperature system when the system contains a sufficiently large number of molecules. From these ideas, we can now find the relationship between the energy values, \({\epsilon }_i\), and the corresponding probabilities,
\[P_i=P\left({\epsilon }_i\right)=g_i\rho \left({\epsilon }_i\right).\nonumber \]
Let us consider the microstates of an isolated system whose energy is \(E^{\#}\). For any population set, \(\{N_1,\ N_2,\dots ,N_i,\dots \}\), that has energy \(E^{\#}\), the following relationships apply.
- The sum of the energy-level populations is the total number of molecules: \[N=N_1+N_2+\dots +N_i=\displaystyle \sum^{\infty }_{j=1}{N_j}\nonumber \]
- The energy of the system is the sum of the energies of its constituent molecules: \[E^{\#}=N_1{\epsilon }_1+N_2{\epsilon }_2+\dots +N_i{\epsilon }_i=\displaystyle \sum^{\infty }_{j=1}{N_j}{\epsilon }_j\nonumber \]
- The product of powers of quantum-state probabilities is a constant: \[{\rho \left({\epsilon }_1\right)}^{N_1}{\rho \left({\epsilon }_1\right)}^{N_1}\dots {\rho \left({\epsilon }_1\right)}^{N_1}\dots =\textrm{ĸ}\nonumber \] or, equivalently, \[\begin{align*} N_1{ \ln \rho \left({\epsilon }_1\right)\ }+N_2{ \ln \rho \left({\epsilon }_2\right)\ }+\dots +N_i{ \ln \rho \left({\epsilon }_i\right)\ }+ … &=\displaystyle \sum^{\infty }_{i=1}{N_i{ \ln \rho \left({\epsilon }_i\right)\ }} \\[4pt] &={ \ln \textrm{ĸ}\ } \end{align*}\nonumber \]
- For the system at constant temperature, the sum of the energy-level probabilities is one. When we infer that the constant-temperature system and the isolated system are functionally equivalent, we assume that this is true also for the isolated system: \[1=P\left({\epsilon }_1\right)+P\left({\epsilon }_2\right)+\dots +P\left({\epsilon }_i\right)+\dots =\displaystyle \sum^{\infty }_{j=1}{P\left({\epsilon }_j\right)}\nonumber \]
We want to find a function, \(\rho \left(\epsilon \right)\), that satisfies all four of these conditions. One way is to keep trying functions that look like they might work until we find one that does. A slightly more sophisticated version of this approach is to try the most general possible version of each such function and see if any set of restrictions will make it work. We could even try an infinite series. Suppose that we are clever (or lucky) enough to try the series solution
\[{ \ln \rho \left(\epsilon \right)\ }=c_0+c_1\epsilon +\dots +c_i{\epsilon }^i+\dots =\displaystyle \sum^{\infty }_{k=0}{c_k}{\epsilon }^k\nonumber \]
Then the third condition becomes
\[\begin{align*} {\ln \textrm{ĸ}\ } &=\displaystyle \sum^{\infty }_{i=1}{N_i}{ \ln \rho \ }\left({\epsilon }_i\right) \\[4pt]&=\displaystyle \sum^{\infty }_{i=1}{N_i}\displaystyle \sum^{\infty }_{k=0}{\left[c_k{\epsilon }^k_i\right]}\\[4pt]&=\displaystyle \sum^{\infty }_{k=0}{\displaystyle \sum^{\infty }_{i=1}{c_kN_i{\epsilon }^k_i}}=c_0\displaystyle \sum^{\infty }_{i=1}{N_i}{\epsilon }^0_i+c_1\displaystyle \sum^{\infty }_{i=1}{N_i}{\epsilon }^1_i+\dots +c_k\displaystyle \sum^{\infty }_{k=2}{\displaystyle \sum^{\infty }_{i=1}{N_i{\epsilon }^k_i}}+\dots \\[4pt]&=c_0N+c_1E^{\#}+\dots +c_k\displaystyle \sum^{\infty }_{k=2}{\displaystyle \sum^{\infty }_{i=1}{N_i{\epsilon }^k_i}}+\dots \end{align*}\nonumber \]
We see that the coefficient of \(c_0\) is \(N\) and the coefficient of \(c_1\) is the total energy, \(E^{\#}\). Therefore, the sum of the first two terms is a constant. We can make the trial function satisfy the third condition if we set \(c_k=0\) for all \(k>1\). We find
\[{ \ln \textrm{ĸ}\ }=\displaystyle \sum^{\infty }_{i=1}{N_i}{ \ln \rho \ }\left({\epsilon }_i\right)=\displaystyle \sum^{\infty }_{i=1}{N_i}\left(c_0+c_1{\epsilon }_i\right)\nonumber \]
The last equality is satisfied if, for each quantum state, we have
\[{ \ln \rho \ }\left({\epsilon }_i\right)=c_0+c_1{\epsilon }_i\nonumber \] or \[\rho \left({\epsilon }_i\right)=\alpha \ \mathrm{exp}\left(c_1{\epsilon }_i\right)\nonumber \]
where \(\alpha =\mathrm{exp}\left(c_0\right)\). Since the \({\epsilon }_i\) are positive and the probabilities \(\rho \left({\epsilon }_i\right)\) lie in the interval \(0<\rho \left({\epsilon }_i\right)<1\), we must have \(c_1<0\). Following custom, we let \(c_1=-\beta\), where \(\beta\) is a constant, and \(\beta >0\). Then,
\[\rho \left({\epsilon }_i\right)=\alpha \ \mathrm{exp}\left(-\beta {\epsilon }_i\right)\nonumber \] and \[P_i=g_i\rho \left({\epsilon }_i\right)=\alpha g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)\nonumber \]
The fourth condition is that the energy-level probabilities sum to one. Using this, we have
\[1=\displaystyle \sum^{\infty }_{i=1}{P\left({\epsilon }_i\right)}=\alpha \displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}\nonumber \]
The sum of exponential terms is so important that it is given a name. It is called the molecular partition function . It is often represented by the letter “\(z\).” Letting
\[z=\displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}\nonumber \] we have
\[\alpha =\frac{1}{\displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}}=z^{-1}\nonumber \]
Thus, we have the Boltzmann probability:
\[\begin{align*} P\left({\epsilon }_i\right) &=g_i\rho \left({\epsilon }_i\right) \\[4pt] &=\frac{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}{\displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}} \\[4pt] &=\frac{g_i}{z}\ \mathrm{exp}\left(-\beta {\epsilon }_i\right) \end{align*} \nonumber \]
The probability of an energy level depends only on its degeneracy, \(g_i\), its energy, \({\epsilon }_i\), and the constant \(\beta\). Since the equilibrium-characterizing population set is determined by the probabilities, we have \(P_i={N^{\textrm{⦁}}_i}/{N}\), and
\[\frac{N^{\textrm{⦁}}_i}{N}=\frac{g_i}{z}\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)\nonumber \]
In Section 21.2 , we develop Lagrange’s method of undetermined multipliers . In Section 21.3 , we develop the same result by applying Lagrange’s method to our model for the probabilities of the microstates of an isolated system. That is, we find the Boltzmann probability equation by applying Lagrange’s method to the entropy relationship,
\[S=-Nk\displaystyle \sum^{\infty }_{i=1}{P_i}{ \ln \rho \left({\epsilon }_i\right)\ }\nonumber \]
that we first develop in § 20-11. In § 4, we find the Boltzmann probability equation by using Lagrange’s method to find the values of \(N^{\textrm{⦁}}_i\) that produce the largest possible value for \(W_{max}\) in an isolated system. This argument requires us to assume that there is a very large number of molecules in each of the occupied energy levels of the most probable population set. Since our other arguments do not assume anything about the magnitude of the various \(N^{\textrm{⦁}}_i\), it is evident that some of the assumptions we make when we apply Lagrange’s method to find the \(N^{\textrm{⦁}}_i\) are not inherent characteristics of our microscopic model.


18.4: Electric field and potential at the surface of a conductor
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If we consider a conducting sphere of radius, \(R\) , with charge, \(+Q\) , the electric field at the surface of the sphere is given by:
\[\begin{aligned} E=k\frac{Q}{R^2}\end{aligned}\]
as we found in the Chapter 17. If we define electric potential to be zero at infinity, then the electric potential at the surface of the sphere is given by:
\[\begin{aligned} V=k\frac{Q}{R}\end{aligned}\]
In particular, the electric field at the surface of the sphere is related to the electric potential at its surface by:
\[\begin{aligned} E=\frac{V}{R}\end{aligned}\]
Thus, if two spheres are at the same electric potential, the one with the smaller radius will have a stronger electric field at its surface.
Because a conducting sphere is symmetric, the charges will distribute themselves symmetrically around the whole outer surface of the sphere. The charge per unit area, \(\sigma\) , at the surface of the sphere is thus given by:
\[\begin{aligned} \sigma &= \frac{Q}{4\pi R^2}\end{aligned}\]
The charge density can be related to the electric field at the surface of the sphere:
\[\begin{aligned} E&=k\frac{Q}{R^2}=k\frac{4\pi R^2\sigma}{R^2}=4\pi\sigma k=\frac{\sigma}{\epsilon_0}\end{aligned}\]
where in the last equality, we used \(k\) with \(\epsilon_0\) and confirmed the general result from Section 17.3, where we determined the electric field near a conductor with surface charge, \(\sigma\) .
Consider a sphere of radius, \(R_1\) , that carries total charge, \(+Q\) . A neutral second, smaller, conducting sphere, of radius \(R_2\) is then connected to the first sphere, using a conducting wire, as in Figure \(\PageIndex{1}\).
Because the charges on the large sphere can move around freely, some of them will move to the smaller sphere. Very quickly, the charges will stop moving and the spheres of radius, \(R_1\) and \(R_2\) , will end up carrying charges, \(Q_1\) and \(Q_2\) , respectively (we assume that the wire is small enough that negligible amounts of charge are distributed on the wire). Since the two conducting spheres are connected by a conductor, they form an equipotential, and are thus at the same voltage, \(V\) , relative to infinity. Since the two spheres are at the same electric potential, the electric field at the surface of each sphere are related:
\[\begin{aligned} E_1&=\frac{V}{R_1}\\[4pt] E_2&=\frac{V}{R_2}\\[4pt] \therefore \frac{E_2}{E_1}&=\frac{R_1}{R_2}\\[4pt] \therefore E_2&=E_1\frac{R_1}{R_2}\end{aligned}\]
and the electric field at the surface of the smaller sphere, \(E_2\) , is stronger since \(R_2<R_1\) . We can also compare the surface charge densities on the two spheres:
\[\begin{aligned} E_1&=\frac{\sigma_1}{\epsilon_0}\\[4pt] E_2&=\frac{\sigma_2}{\epsilon_0}\\[4pt] \therefore \frac{\sigma_2}{\sigma_1}&=\frac{E_2}{E_1}=\frac{R_1}{R_2}\\[4pt] \therefore \sigma_2&=\sigma_1 \frac{R_1}{R_2}\end{aligned}\]
and we find that the charge density is higher on the smaller sphere. Thus, there are more charges per unit area on the smaller sphere than the bigger sphere.
We can generalize this model to describe charges on any charged conducting object. If charges are deposited on a conducting object that is not a sphere, as in Figure \(\PageIndex{2}\), they will not distribute themselves uniformly. Instead, there will be a higher charge density (charges per unit area), near parts of the object that have a small radius of curvature (sharp points on the object in particular), just as the charge density was higher on the smaller sphere described above. As a consequence of the higher concentration of charges near the “pointier” parts of the object, the electric field at the surface will be the strongest in those regions (as it is stronger at the surface of the smaller sphere described above).
In air, if the electric field exceeds a magnitude of approximately \(3\times 10^{6}\text{V/m}\) , the air is said to ”electrically breakdown”. The strong electric field can remove electron from atoms in the air, ionizing the air in a chain reaction and making it conductive. Thus, if the electric field at a point on the surface of a conductor is very strong, the air near that point will break down, and charges will leave the conductor, through the air, to find a location with lower electric potential energy (usually the ground). Electric breakdown is what we experience as a spark (or lightning, on a larger scale), and is usually a discrete (and potentially dramatic) event. Corona discharge is another mechanism whereby the strong electric field can make the air conductive, but in this case charges leak into the air more gradually, unlike in the case of electrical break down. Charges leaking into air through Corona discharge will emit a faint blueish light (the “Corona”) as well as an audible hissing sound.
Objects that are designed to hold a high electric potential (for example the electrodes on high voltage lines) are usually made very carefully so that they have a very smooth surface and no sharp edges. This reduces the risk of breakdown or corona discharge at the surface which would result in a loss of charge.
Contrary to popular belief, lightning rods are not designed to attract lightening. Instead, lightning rods are designed to be conductors with a very sharp point, so that corona discharge can occur at their tip. This allows charges to slowly leak off from the Earth into the cloud through Corona discharge, thereby reducing the potential difference between the cloud and Earth so that a lightning strike (electrical breakdown) does not occur. When a lightning strike does occur, it will hit the lightning rod, since the electric field at the top of the rod is high and that is the most likely point for the air to break down; but, that is not the goal of the lightning rod!


12.4: Solving Equivalent Ratio Problems
Lesson
Let's practice getting information from our partner.
Exercise \(\PageIndex{1}\): What Do You Want to Know?
Consider the problem: A red car and a blue car enter the highway at the same time and travel at a constant speed. How far apart are they after 4 hours?
What information would you need to be able to solve the problem?
Exercise \(\PageIndex{2}\): Info Gap: Hot Chocolate and Potatoes
Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.
If your teacher gives you the problem card :
- Silently read your card and think about what information you need to be able to answer the question.
- Ask your partner for the specific information that you need.
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Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently.
- Read the data card and discuss your reasoning.
If your teacher gives you the data card :
- Silently read your card.
-
Ask your partner
“What specific information do you need?”
and wait for them to
ask
for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
- Read the problem card and solve the problem independently.
- Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Exercise \(\PageIndex{3}\): Comparing Reading Rates
- Lin read the first 54 pages from a 270-page book in the last 3 days.
- Diego read the first 100 pages from a 320-page book in the last 4 days.
- Elena read the first 160 pages from a 480-page book in the last 5 days.
If they continue to read every day at these rates, who will finish first, second, and third? Explain or show your reasoning.
Are you ready for more?
The ratio of cats to dogs in a room is \(2:3\). Five more cats enter the room, and then the ratio of cats to dogs is \(9:11\). How many cats and dogs were in the room to begin with?
Summary
To solve problems about something happening at the same rate, we often need:
- Two pieces of information that allow us to write a ratio that describes the situation.
- A third piece of information that gives us one number of an equivalent ratio. Solving the problem often involves finding the other number in the equivalent ratio.
Suppose we are making a large batch of fizzy juice and the recipe says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” We know that the ratio of cranberry juice to soda water is \(5:2\), and that we need 2.5 cups of cranberry juice per cup of soda water.
We still need to know something about the size of the large batch. If we use 16 cups of soda water, what number goes with 16 to make a ratio that is equivalent to \(5:2\)?
To make this large batch taste the same as the original recipe, we would need to use 40 cups of cranberry juice.
| cranberry juice (cups) | soda water (cups) |
|---|---|
| \(5\) | \(2\) |
| \(2.5\) | \(1\) |
| \(40\) | \(16\) |
Glossary Entries
Definition: Table
A table organizes information into horizontal rows and vertical columns . The first row or column usually tells what the numbers represent.
For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.
| pet | tail length (inches) |
|---|---|
| dog | \(22\) |
| cat | \(12\) |
| mouse | \(2\) |
Practice
Exercise \(\PageIndex{4}\)
A chef is making pickles. He needs 15 gallons of vinegar. The store sells 2 gallons of vinegar for $3.00 and allows customers to buy any amount of vinegar. Decide whether each of the following ratios correctly represents the price of vinegar.
- 4 gallons to $3.00
- 1 gallon to $1.50
- 30 gallons to $45.00
- $2.00 to 30 gallons
- $1.00 to \(\frac{2}{3}\) gallon
Exercise \(\PageIndex{5}\)
A caterer needs to buy 21 pounds of pasta to cater a wedding. At a local store, 8 pounds of pasta cost $12. How much will the caterer pay for the pasta there?
- Write a ratio for the given information about the cost of pasta.
- Would it be more helpful to write an equivalent ratio with 1 pound of pasta as one of the numbers, or with $1 as one of the numbers? Explain your reasoning, and then write that equivalent ratio.
- Find the answer and explain or show your reasoning.
Exercise \(\PageIndex{6}\)
Lin is reading a 47-page book. She read the first 20 pages in 35 minutes.
- If she continues to read at the same rate, will she be able to complete this book in under 1 hour?
- If so, how much time will she have left? If not, how much more time is needed? Explain or show your reasoning.
Exercise \(\PageIndex{7}\)
Diego can type 140 words in 4 minutes.
- At this rate, how long will it take him to type 385 words?
- How many words can he type in 15 minutes?
If you get stuck, consider creating a table.
Exercise \(\PageIndex{8}\)
A train that travels 30 miles in \(\frac{1}{3}\) hour at a constant speed is going faster than a train that travels 20 miles in \(\frac{1}{2}\) hour at a constant speed. Explain or show why.
(From Unit 2.3.5)
Exercise \(\PageIndex{9}\)
Find the surface area of the polyhedron that can be assembled from this net. Show your reasoning.
(From Unit 1.5.3)


10.8: Notes
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- Huan Piñón
- University of California Press
1 Toby Miller and Marie C. Leger, “Runaway Production, Runaway Consumption, Runaway Citizenship: The New International Division of Cultural Labor,” Emergences 11.1 (2001): 89–115.
2 David Hesmondhalgh, Cultural Industries , 2nd ed. (Los Angeles, CA: Sage, 2006).
3 In the last five years there has been a reformulation of telecommunication laws in the main domestic markets across the region with respect to quotas on the distribution of national and independently produced content with economic incentives for independent productions; however, the results of such legislation are uneven.
4 Toby Miller, chapter 2 in this volume.
5 Michael Curtin and Kevin Sanson, chapter 1 in this volume.
6 Petr Szczepanik, chapter 7 in this volume.
7 Michael Keane, chapter 16 in this volume.
8 Joseph Straubhaar, “Beyond Media Imperialism: Asymmetrical Interdependency and Cultural Proximity,” Critical Studies in Mass Communication 8 (1991): 39–59.
9 Among the most visible indies are Argos, Adicta Films, El Mall, Lemon, and Canana Film, Blind Spot in Mexico; RTI, Teleset, Vista Producciones, FoxTelecolombia, Laberinto Producciones, BETV in Colombia; Cuatro Cabezas, Underground, Ideas del Sur, RGB, Chris Morena, Pol-ka, and Endemol in Argentina; Del Barrio Producciones, Imizu, Teatro Libre, and Sol Entertainment Producciones in Perú; and Bueno Puerto Producciones, Valcine Producciones, Wood Producciones, and My Friend Entertainment Producciones in Chile. Conspiração Filmes, HBO O 2 Filmes, and Mixer Brazil represent the most visible examples of a much larger group of indies.
10 The recognition of the Latin American television context is the product of years of research as part of the Ibero-American Observatory of Television Fiction (Obitel); but this chapter is based on ethnographic work done at NATPE in summer 2012, and a handful of interviews done in Fall 2014 with high executives and above-the-line creative personnel from independent production houses: Blind Spot and Argos (Mexico), Laberinto Producciones (Colombia), Del Barrio Producciones (Peru), and MicroTime (Uruguay). The pool of interviewees from independent production houses is composed of six top executives. Their shared characteristics are active involvement in the production of television programming for television networks in their countries as well as involvement in coproduction or commissioned productions, particularly fiction, with a U.S. or global media conglomerate.
11 Jesús Martín-Barbero, “Memory and Form in the Latin American Soap Opera,” in The Television Studies Reader , ed. Robert Allen & Annette Hill (London: Routledge, 2004).
12 James Schwoch, The American Radio Industry and Its Latin American Activities , 1900–1939 (Urbana: University of Illinois Press, 1990).
13 Noreene Janus, “Advertising and the Mass Media in the Era of the Global Corporation,” in Communication and Social Structure: Critical Studies in Mass Media Research , ed. Emile McAnany, Jorge Schnitman, and Noreene Janus (New York: Praeger, 1981).
14 John Sinclair and Joseph Straubhaar, Latin American Television Industries (London: British Film Institute, 2013).
15 Elizabeth Fox, Latin American Broadcasting: From Tango to Telenovela (Bedfordshire: University of Luton and John Libbey Media, 1997); Elizabeth Fox and Silvio Waisbord, eds., Latin Politics, Global Media (Austin: University of Texas Press, 2002).
16 Joseph Straubhaar, World Television from Global to Local (Los Angeles: Sage, 2007).
17 Joseph Straubhaar, “Beyond Media Imperialism: Asymmetrical Interdependency and Cultural Proximity,” Critical Studies in Mass Communicatio n 8 (1991): 39–59.
18 Guillermo Orozco, ed., Historias de la televisión en América Latina (Barcelona: Gedisa, 2003).
19 Ibid.
20 Nora Mazziotti, La industria de la telenovela: La producción de ficción en América Latina (Buenos Aires: Paidós, 1996).
21 Raúl Trejo, “Muchos medios en pocas manos: Concentración televisiva y democracia en América Latina,” Revista Brasileira de ciencias da comunicacao 33.1 (2010): 17–51.
22 For instance in Mexico, TV Azteca sued Alan Tatcher for working with indie Nostromo in a new reality show for Telemundo in 2006; TV Azteca also sued talent working with indie Argos in a series for Cadena Tres in 2013.
23 There were five in Argentina, six in Brazil, seven in Chile, five in Colombia, seven in Ecuador, five in Mexico, six in Peru, four in Uruguay and fourteen in Venezuela. Guillermo Orozco and María I. Vasallo, eds., Transmedia Production Strategies in Television Fiction (Porto Alegre, Brazil: Globo Comunicação e Participações and Sulina Editora, 2014).
24 In Mexico, Televisa and TV Azteca hold 95 percent of the audience share; in Colombia, RCN and Caracol TV hold 97 percent; in Venezuela, Venevision and Televen 81 percent; in Brazil, TV Globo holds 40 percent; in Argentina, El Trece and Telefe hold 60 percent; in Peru, America TV, ATV and Frecuencia Latina hold 86 percent; and in Chile Canal 13, Chilevision, and TVN hold 70 percent of the audience share. Ibid.
25 LAMAC, “Penetracion de TV de paga,” Latin American Multichannel Advertising Council , www.lamac.org/.
26 Interview with executive producer from Laberinto Producciones, Colombia, November 25, 2014.
27 Interview with executive producer from Argos, Mexico, January 29, 2013.
28 Interview with artistic director from Del Barrio Producciones, Peru, November 27, 2014.
29 Interview with executive producer from Blind Spot, Mexico, November 19, 2014.
30 Telemundo kept the distribution rights in the United States, Canada, and Puerto Rico. Mary Sutter, “Telemundo, Argos to Link,” Variety , October 13, 2000, http://variety.com/2000/tv/news/tele...nk-1117787691/ .
31 Joe Flint, “Productora Mexicana dará telenovelas a Telemundo,” Mural , October 2000, 5.
32 Interview with screenwriter for Argos, Telemundo, TV Azteca, HBO, Cadena Tres, August 22, 2006.
33 Nikolas Maksymiv, “Imagen y Argos Anuncian una Alianza,” Noticias financieras (Miami), January 21, 2010, ProQuest (466639295).
34 Interview with screenwriter for Argos, Telemundo, TV Azteca, HBO, and Cadena Tres, November 26, 2014.
35 Interview with executive producer from Argos, Mexico, January 29, 2013.
36 Juan Piñón, “Reglocalization and the Rise of the Network Cities System in Producing Telenovelas for Hemispheric Audiences,” Journal of International Cultural Studies 17.6 (2014): 655–671.
37 Sara Bibel, “Telemundo’s ‘El Señor de los Cielos’ Wins First-Ever International Emmy for Non-English Language U.S. Primetime Program,” TVbythenumbers , November 25, 2014, http://tvbythenumbers.zap2it.com/201...rogram/332190/ .
38 Hernán Casciari, “HBO con acento latinos,” El país , February 2009, http://blogs.elpais.com/espoiler/200...to-latino.html .
39 Interview with screenwriter for Argos, Telemundo, TV Azteca, HBO, and Cadena Tres, November 26, 2014.
40 Ibid.
41 Marie A. De la Fuente, “Sony Will Sell Telenovelas at Mip,” Variety , March 29, 2009, http://variety.com/2009/film/news/so...ip-1118001742/ .
42 Interview with executive producer from Laberinto Producciones, Colombia, November 25, 2014.
43 Interview with artistic director from Del Barrio Producciones, Peru, November 27, 2014.
44 Interview with executive producer from Microtime, Uruguay, December 1, 2014.
45 Interview with screenwriter for Argos, Telemundo, TV Azteca, HBO, and Cadena Tres, November 26, 2014.
46 Interview with artistic director from Del Barrio Producciones, Peru, November 27, 2014.
47 Pierre Bourdieu, The Field of Cultural Production: Essays on Art and Literature (New York: Columbia University Press, 1993).


9.7: Embedding Summative ADAPT Assessments into Textbook
Section 5.7 of the Construction guide discusses how to embed interactive assessment into Libretexts pages in a formative manner (i.e., without a gradebook). The ADAPT system also allows the embedding of Adapt assessments in your LibreTexts book in a summative manner. This allows students to find them easier and complete the assignments without leaving the textbook.
You can create a separate page for embedding assignments or add them to existing pages. See Construction Guide to learn more on page creation.
Note
Students have to be logged into their Adapt account to see the embedded assignment. Assignments won’t be visible in the print version of your book.
Embedding Summative Assessments
To embed a new Adapt assignment to your LibreTexts page:
-
First add the assignment and assessments to ADAPT itself. See Adding/Importing Assignments and Adding New Assessments to learn more
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Copy the assignment ID. You can select the text → Right click → Copy or just click on the copy icon next to the ID
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Open the LibreText page that you’d like to add this assessment to
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Click Edit (See more on page editing in the Basic Editing section of the Construction Guide)
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Select Adapt Icon to insert a new Adapt assessment
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Type/paste your Adapt ID that you copied in Step 2 and click “OK”
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You can edit the numbering if you have more than one embedded assessment. If you have 3 Adapt assessments on the same page by default they all will be labeled “ADAPT 1”, but if you change the numbering you can have “ADAPT 1”, “ADAPT 2” and “ADAPT 3”
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Click “Save” to save your page. You should see your Adapt assessment embedded.
Note
If you see an error message “You are not logged in!” you need to log in using Campus Login or (if you don’t have campus login setup) you need to open Adapt website, log into your account and then refresh the LibreText page.


7.3.5: Other Acellular Infectious Agents- Viroids and Prions
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Define viroid and name an infection caused by a viroid.
- Define prion and name 3 protein misfolding diseases that apprear to be initiated by prions.
Viroids and Prions
Viroids are even more simple than viruses. They are small, circular, single-stranded molecules of infectious RNA lacking even a protein coat. They are the cause of a few plant diseases such as potato spindle-tuber disease,cucumber pale fruit, citrus exocortis disease, and cadang-cadang (coconuts).
Prions are infectious protein particles responsible for a group of transmissible and/or inherited neurodegenerative diseases including Creutzfeldt-Jakob disease, kuru, and Gerstmann-Straussler-syndrome in humans, as well as scrapie in sheep and goats, and bovine spongiform encephalopathy (mad cow disease) in cattle and in humans (where it is called new variant Creutzfeldt–Jakob disease humans). The infections are often referred to as transmissible spongiform encephalopathies.
Most evidence indicates that the infectious prion proteins are modified (misfolded) forms of normal proteins coded for by a host gene in the brain. It is thought that the normal prion protein, expressed on stem cells in the bone marrow and on cells that will become neurons, plays a role in the maturation of neurons. In the case of the disease scrapie, the normal prion protein in an animal without the disease has alpha-helices in the proteins secondary structure (Figure \(\PageIndex{1}\)) while the scrapie prion protein in diseased animals has beta-sheets for the secondary structure (Figure \(\PageIndex{2}\)). When the scrapie prion protein contacts the normal protein it causes it to change its configuration to the scrapie beta-sheet form. This suggests that the conversion of a normal prion protein into an infectious prion protein may be catalyzed by the prion protein itself upon entering the brain. Inherited forms may be a result of point mutations that make the prion protein more susceptible to a change in its protein structure.
There is growing evidence that other probable protein misfolding diseases initiated by prions include Alzheimer's disease, Hunington's disease, Parkinson's disease, frontotemporal dementias, amyotrophic lateral sclerosis, and certain cancers.
Summary
- Viroids are small, circular, single-stranded molecules of infectious RNA that cause several plant diseases.
- Prions are infectious protein particles responsible for a group of transmissible and/or inherited neurodegenerative diseases as a result of prion protein misfolding.
- Diseases including Creutzfeldt-Jakob disease Gerstmann-Straussler-syndrome, and mad cow disease.
- There is growing evidence that other probable protein misfolding diseases initiated by prions include Alzheimer's disease, Hunington's disease, Parkinson's disease, amyotrophic lateral sclerosis, and certain cancers.


15.5: Assignment- It’s a Small Ad After All
It’s a Small Ad After All …
The process of conceptualizing and then conveying a final product or service worldwide with the hopes of reaching the international marketing community. Proper global marketing has the ability to catapult a company to the next level, if they do it correctly. Different strategies are implemented based on the region the company is marketing to. For example, the menu at McDonald’s varies based on the location of the restaurant. The company focuses on marketing popular items within the country. Global marketing is especially important to companies that provide products or services that have a universal demand such as automobiles and food.
Read more: http://www.businessdictionary.com/definition/global-marketing.html#ixzz2mMy3Shto
In order to complete this discussion board assignment , you will need to complete the following:
- Using the Internet, find an ONE example of “Global Marketing” that has been unsuccessful. Be sure to copy links to your articles or sources so that you can put them into your initial post. Your fellow classmates should have the opportunity to read up on your example!
-
In your
INITIAL POST
you will:
- Provide a brief synopsis of the example that you selected. Be sure to include the company, market they were entering and why you considered it to be a failure.
- Why do you think the company took the approach it did in marketing to an international market?
- What did they NOT consider in marketing their product, good or service?
- What consequences did the company suffer as a result of this failed marketing effort?
- Respond to at least TWO of your classmates posts before the deadline.
Goal: Identify the various strategies of international business and the strategic ways to enter a foreign market.
- International Business. Authored by : Linda Williams. Provided by : Tidewater Community College. Located at : http://www.tcc.edu . Project : Z Degree Project. License : CC BY: Attribution


1.8: Projection Matrices
The two-by-two projection matrix projects a vector onto a specified vector in the \(x\)-\(y\) plane. Let \(\mathbf{u}\) be a unit vector in \(\mathbb{R}^2\). The projection of an arbitrary vector \(\mathbf{x} = \langle x_1, x_2\rangle\) onto the vector \(\mathbf{u} = \langle u_1, u_2\rangle\) is determined from
\[\text{Proj}_{\mathbf{u}}(\mathbf{x})=(\mathbf{x}\cdot\mathbf{u})\mathbf{u}=(x_1u_1+x_2u_2)\langle u_1,u_2\rangle .\nonumber \]
In matrix form, this becomes
\[\left(\begin{array}{c}p_1\\p_2\end{array}\right)=\left(\begin{array}{cc}u_1^2&u_1u_2 \\ u_1u_2&u_2^2\end{array}\right)\left(\begin{array}{c}x_1\\x_2\end{array}\right).\nonumber \]
The projection matrix \(\text{P}_{\mathbf{u}}\), then, can be defined as
\[\begin{aligned}\text{P}_{\mathbf{u}}&=\left(\begin{array}{cc}u_1^2&u_1u_2\\u_1u_2&u_2^2\end{array}\right) \\ &=\left(\begin{array}{c}u_1\\u_2\end{array}\right)\left(\begin{array}{cc}u_1&u_2\end{array}\right) \\ &=\text{uu}^{\text{T}},\end{aligned} \nonumber \]
which is an outer product. Notice that \(\text{P}_{\mathbf{u}}\) is symmetric.
Show that \(\text{P}_{\mathbf{u}}^2=\text{P}_{\mathbf{u}}\).
Solution
It should be obvious that two projections is the same as one. To prove, we have
\[\begin{aligned}\text{P}_{\mathbf{u}}^2&=(\text{uu}^{\text{T}})(\text{uu}^{\text{T}}) \\ &=\text{u}(\text{u}^{\text{T}}\text{u})\text{u}^{\text{T}}&\quad\text{(associative law)} \\ &=\text{uu}^{\text{T}}&\quad (\mathbf{u}\text{ is a unit vector)} \\ &=\text{P}_{\mathbf{u}}.\end{aligned} \nonumber \]