Chapter 9

Match the word or phrase with the most appropriate choice from the column on the right. _______Total cost a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Classify each of the following statements as either true or false. A square matrix has the same number of rows and columns.

Solve. The sum of three numbers is \(57 .\) The second is 3 more than the first. The third is 6 more than the first. Find the numbers.

Match the word or phrase with the most appropriate choice from the column on the right. _______Total revenue a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Classify each of the following statements as either true or false. \(A 3 \times 4\) matrix has 3 rows and 4 columns.

Solve. The sum of three numbers is \(5 .\) The first number minus the second plus the third is \(1 .\) The first minus the third is 3 more than the second. Find the numbers.

Match the word or phrase with the most appropriate choice from the column on the right. _______Total profit a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Classify each of the following statements as either true or false. A determinant is a number.

Complete each of the following statements. Each number in a matrix is called a(n)_____ or element.

Solve. The sum of three numbers is \(26 .\) Twice the first minus the second is 2 less than the third. The third is the second minus three times the first. Find the numbers.

Match the word or phrase with the most appropriate choice from the column on the right. _______Fixed costs a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Complete each of the following statements. The plural of the word matrix is _____.

Solve. The sum of three numbers is \(105 .\) The third is 11 less than ten times the second. Twice the first is 7 more than three times the second. Find the numbers.

Classify each of the following statements as either true or false. If, when we are solving a system of three equations, a false equation results from adding a multiple of one equation to another, the system is inconsistent.

Match the word or phrase with the most appropriate choice from the column on the right. _______Variable costs a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Solve. Geometry. In triangle \(A B C,\) the measure of angle \(B\) is three times that of angle \(A .\) The measure of angle \(C\) is \(20^{\circ}\) more than that of angle \(A .\) Find the angle measures.

Classify each of the following statements as either true or false. If, when we are solving a system of three equations, an identity results from adding a multiple of one equation to another, the equations are dependent.

Match the word or phrase with the most appropriate choice from the column on the right. _______Break-even point a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Classify each of the following statements as either true or false. Whenever Cramer's rule yields a numerator that is 0 , the equations are dependent.

Solve. Geometry: In triangle \(A B C\), the measure of angle \(B\) is twice the measure of angle \(A .\) The measure of angle \(C\) is \(80^{\circ}\) more than that of angle \(A .\) Find the angle measures.

Match the word or phrase with the most appropriate choice from the column on the right. _______Equilibrium point a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Evaluate. $$ \left|\begin{array}{ll} {5} & {1} \\ {2} & {4} \end{array}\right| $$

Solve using matrices. $$ \begin{aligned} x+2 y &=11 \\ 3 x-y &=5 \end{aligned} $$

Solve. Scholastic Aptitude Test. Many high-school students take the Scholastic Aptitude Test (SAT). Beginning in March \(2005,\) students taking the SAT received three scores: a critical reading score, a mathematics score, and a writing score. The average total score of 2009 high-school seniors who took the SAT was \(1509 .\) The average mathematics score exceeded the reading score by 14 points and the average writing score was 8 points less than the reading score. What was the average score for each category?

Determine whether \((2,-1,-2)\) is a solution of the system $$\begin{aligned} x+y-2 z &=5 \\ 2 x-y-z &=7 \\ -x-2 y-3 z &=6 \end{aligned}$$

Match the word or phrase with the most appropriate choice from the column on the right. _______Price a) The amount of money that a company takes in b) The sum of fixed costs and variable costs c) The point at which total revenue equals total cost d) What consumers pay per item e) The difference between total revenue and total cost f) What companies spend whether or not a product is produced g) The point at which supply equals demand h) The costs that vary according to the number of items produced

Evaluate. $$ \left|\begin{array}{rr} 3 & 2 \\ 2 & -3 \end{array}\right| $$

Solve using matrices. $$ \begin{aligned} x+3 y &=16 \\ 6 x+y &=11 \end{aligned} $$

Advertising. In 2008 , U.S. companies spent a total of S1 18.2 billion on newspaper, television, and magazine ads. The total amount spent on television ads was \(\$ 10.8\) billion more than the amount spent on newspaper and magazine ads together. The amount spent on magazine ads was \(\$ 3.5\) billion more than the amount spent on newspaper ads. How much was spent on each form of advertising?

Determine whether \((-1,-3,2)\) is a solution of the system $$\begin{array}{r} {x-y+z=4} \\ {x-2 y-z=3} \\ {3 x+2 y-z=1} \end{array}$$

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{array}{l} {C(x)=45 x+300,000} \\ {R(x)=65 x} \end{array} $$

Evaluate. $$ \left|\begin{array}{rr} 10 & 8 \\ -5 & -9 \end{array}\right| $$

Solve using matrices. $$ \begin{array}{r} {x+4 y=8} \\ {3 x+5 y=3} \end{array} $$

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{array}{l} {C(x)=25 x+270,000} \\ {R(x)=70 x} \end{array} $$

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} x+y-z &=0 \\ 2 x-y+z &=3 \\ -x+5 y-3 z &=2 \end{aligned}$$

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{aligned} &C(x)=15 x+3100\\\ &R(x)=40 x \end{aligned} $$

Solve using matrices. $$ \begin{aligned} &6 x-2 y=4\\\ &7 x+y=13 \end{aligned} $$

Automobile Pricing. The basic model of a 2010 Honda Civic Hybrid with a car cover cost \(\$ 24,030 .\) When equipped with satellite radio and a car cover, the vehicle's price rose to \(\$ 24.340 .\) The price of the basic model with satellite radio was \(\$ 24,110 .\) Find the basic price, the price of satellite radio, and the price of a car cover.

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} x-y-z &=1 \\ 2 x+y+2 z &=4 \\ x+y+3 z &=5 \end{aligned}$$

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{array}{l} {C(x)=30 x+49,500} \\ {R(x)=85 x} \end{array} $$

Evaluate. $$ \left|\begin{array}{rrr} 2 & 4 & -2 \\ 1 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right| $$

Solve using matrices. $$ \begin{aligned} 3 x+4 y &=7 \\ -5 x+2 y &=10 \end{aligned} $$

Telemarketing. Sven, Laurie, and Isaiah can process 740 telephone orders per day. Sven and Laurie together can process 470 orders, while Laurie and Isaiah together can process 520 orders per day. How many orders can each person process alone?

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} x+y-3 z &=4 \\ 2 x+3 y+z &=6 \\ 2 x-y+z &=-14 \end{aligned}$$

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{array}{l} {C(x)=40 x+22,500} \\ {R(x)=85 x} \end{array} $$

Evaluate. $$ \left|\begin{array}{rrr} -4 & -2 & 3 \\ -3 & 1 & 2 \\ 3 & 4 & -2 \end{array}\right| $$

Solve using matrices. $$ \begin{aligned} 3 x+2 y+2 z &=3 \\ x+2 y-z &=5 \\ 2 x-4 y+z &=0 \end{aligned} $$

Coffee Prices. Reba works at a Starbucks "coffee shop where a \(12-\) oz cup of coffee costs \(\$ 1.65,\) a 16 -oz cup costs \(\$ 1.85,\) and a 20 -oz cup costs \(\$ 1.95 .\) During one busy period, Reba served 55 cups of coffee, emptying six \(144-\mathrm{oz}\) "brewers" while collecting a total of \(\$ 99.65 .\) How many cups of each size did Reba fill?

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} 3 x+4 y-3 z &=4 \\ 5 x-y+2 z &=3 \\ x+2 y-z &=-2 \end{aligned}$$

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{aligned} &C(x)=20 x+10,000\\\ &R(x)=100 x \end{aligned} $$