Intermediate Algebra

Elayn Martin-Gay

Chapter 4

Systems and Equations - all with Video Answers

Educators

Section 1

Solving Systems of Linear Equations in Two Variables

Problem 1

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{aligned}
x-y &=3 \\
2 x-4 y &=8
\end{aligned} \quad(2,-1)\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 2

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{aligned}
x-y &=-4 \\
2 x+10 y &=4
\end{aligned} \quad(-3,1)\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 3

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{array}{ll}
{2 x-3 y=-9} & {(3,5)} \\
{4 x+2 y=-2} \end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 4

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{array}{ll}
{2 x-5 y=-2} & {(4,2)} \\
{3 x+4 y=4} & \end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 5

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{array}{l}
{y=-5 x} \\
{x=-2}
\end{array} \quad(-2,10)\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 6

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{array}{l}
{y=6} \\
{x=-2 y}
\end{array} \quad(-12,6)\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 7

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{aligned}
3 x+7 y &=-19 \\
-6 x &=5 y+8
\end{aligned} \quad\left(\frac{2}{3},-3\right)\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 8

Determine whether each given ordered pair is a solution of each system. See Example 1.
$$
\left\{\begin{aligned}
4 x+5 y &=-7 \\
-8 x &=3 y-1
\end{aligned}\left(\frac{3}{4},-2\right)\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 9

Solve each system by graphing. See Examples 2, through 4.
$$
\left\{\begin{array}{r}
{x+y=1} \\
{x-2 y=4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 10

Solve each system by graphing. See Examples 2, through 4.
$$
\left\{\begin{array}{l}
{2 x-y=8} \\
{x+3 y=11}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 11

Solve each system by graphing. See Examples 2, through 4.
$$
\left\{\begin{array}{r}
{2 y-4 x=0} \\
{x+2 y=5}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 12

Solve each system by graphing. See Examples 2, through 4.
$$
\left\{\begin{array}{r}
{4 x-y=6} \\
{x-y=0}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 13

Solve each system by graphing. See Examples 2, through 4.
$$
\left\{\begin{array}{l}
{3 x-y=4} \\
{6 x-2 y=4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 14

Solve each system by graphing. See Examples 2, through 4.
$$
\left\{\begin{array}{l}
{-x+3 y=6} \\
{3 x-9 y=9}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 15

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{r}
{x+y=10} \\
{y=4 x}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 16

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{aligned}
5 x+2 y &=-17 \\
x &=3 y
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 17

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{l}
{4 x-y=9} \\
{2 x+3 y=-27}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 18

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{c}
{3 x-y=6} \\
{-4 x+2 y=-8}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 19

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{l}
{\frac{1}{2} x+\frac{3}{4} y=-\frac{1}{4}} \\
{\frac{3}{4} x-\frac{1}{4} y=1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 20

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{c}
{\frac{2}{5} x+\frac{1}{5} y=-1} \\
{x+\frac{2}{5} y=-\frac{8}{5}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 21

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{c}
{\frac{x}{3}+y=\frac{4}{3}} \\
{-x+2 y=11}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 22

Solve each system of equations by the substitution method. See Examples 5 and 6.
$$
\left\{\begin{array}{l}
{\frac{x}{8}-\frac{y}{2}=1} \\
{\frac{x}{3}-y=2}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 23

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{c}
{-x+2 y=0} \\
{x+2 y=5}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 24

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{r}
{-2 x+3 y=0} \\
{2 x+6 y=3}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 25

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{r}
{5 x+2 y=1} \\
{x-3 y=7}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 26

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{l}
{6 x-y=-5} \\
{4 x-2 y=6}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 27

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{c}
{\frac{3}{4} x+\frac{5}{2} y=11} \\
{\frac{1}{16} x-\frac{3}{4} y=-1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 28

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{l}
{\frac{2}{3} x+\frac{1}{4} y=-\frac{3}{2}} \\
{\frac{1}{2} x-\frac{1}{4} y=-2}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 29

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{l}
{3 x-5 y=11} \\
{2 x-6 y=2}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 30

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{l}
{6 x-3 y=-3} \\
{4 x+5 y=-9}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 31

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{r}
{x-2 y=4} \\
{2 x-4 y=4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 32

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{array}{l}
{-x+3 y=6} \\
{3 x-9 y=9}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 33

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{aligned}
3 x+y &=1 \\
2 y &=2-6 x
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 34

Solve each system of equations by the elimination method. See Examples 7 through 10.
$$
\left\{\begin{aligned}
y &=2 x-5 \\
8 x-4 y &=20
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 35

Solve each system of equations.
$$
\left\{\begin{array}{l}
{2 x+5 y=8} \\
{6 x+y=10}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 36

Solve each system of equations.
$$
\left\{\begin{aligned}
x-4 y &=-5 \\
-3 x-8 y &=0
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 37

Solve each system of equations.
$$
\left\{\begin{array}{r}
{2 x+3 y=1} \\
{x-2 y=4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 38

Solve each system of equations.
$$
\left\{\begin{array}{r}
{-2 x+y=-8} \\
{x+3 y=11}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 39

Solve each system of equations.
$$
\left\{\begin{array}{c}
{\frac{1}{3} x+y=\frac{4}{3}} \\
{-\frac{1}{4} x-\frac{1}{2} y=-\frac{1}{4}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 40

Solve each system of equations.
$$
\left\{\begin{array}{r}
{\frac{3}{4} x-\frac{1}{2} y=-\frac{1}{2}} \\
{x+y=-\frac{3}{2}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 41

Solve each system of equations.
$$
\left\{\begin{array}{l}
{2 x+6 y=8} \\
{3 x+9 y=12}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 42

Solve each system of equations.
$$
\left\{\begin{aligned}
x &=3 y-1 \\
2 x-6 y &=-2
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 43

Solve each system of equations.
$$
\left\{\begin{array}{c}
{4 x+2 y=5} \\
{2 x+y=-1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 44

Solve each system of equations.
$$
\left\{\begin{array}{l}
{3 x+6 y=15} \\
{2 x+4 y=3}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 45

Solve each system of equations.
$$
\left\{\begin{aligned}
10 y-2 x &=1 \\
5 y &=4-6 x
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 46

Solve each system of equations.
$$
\left\{\begin{array}{r}
{3 x+4 y=0} \\
{7 x=3 y}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 47

Solve each system of equations.
$$
\left\{\begin{array}{r}
{5 x-2 y=27} \\
{-3 x+5 y=18}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 48

Solve each system of equations.
$$
\left\{\begin{array}{l}
{3 x+4 y=2} \\
{2 x+5 y=-1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 49

Solve each system of equations.
$$
\left\{\begin{aligned}
x &=3 y+2 \\
5 x-15 y &=10
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 50

Solve each system of equations.
$$
\left\{\begin{aligned}
y &=\frac{1}{7} x+3 \\
x-7 y &=-21
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 51

Solve each system of equations.
$$
\left\{\begin{aligned}
2 x-y &=-1 \\
y &=-2 x
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 52

Solve each system of equations.
$$
\left\{\begin{array}{r}
{x=\frac{1}{5} y} \\
{x-y=-4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 53

Solve each system of equations.
$$
\left\{\begin{array}{c}
{2 x=6} \\
{y=5-x}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 54

Solve each system of equations.
$$
\left\{\begin{array}{c}
{x=3 y+4} \\
{-y=5}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 55

Solve each system of equations.
$$
\left\{\begin{array}{c}
{\frac{x+5}{2}=\frac{6-4 y}{3}} \\
{\frac{3 x}{5}=\frac{21-7 y}{10}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 56

Solve each system of equations.
$$
\left\{\begin{array}{l}
{\frac{y}{5}=\frac{8-x}{2}} \\
{x=\frac{2 y-8}{3}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 57

Solve each system of equations.
$$
\left\{\begin{array}{r}
{4 x-7 y=7} \\
{12 x-21 y=24}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 58

Solve each system of equations.
$$
\left\{\begin{aligned}
2 x-5 y &=12 \\
-4 x+10 y &=20
\end{aligned}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 59

Solve each system of equations.
$$
\left\{\begin{array}{c}
{\frac{2}{3} x-\frac{3}{4} y=-1} \\
{-\frac{1}{6} x+\frac{3}{8} y=1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 60

Solve each system of equations.
$$
\left\{\begin{array}{l}
{\frac{1}{2} x-\frac{1}{3} y=-3} \\
{\frac{1}{8} x+\frac{1}{6} y=0}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 61

Solve each system of equations.
$$
\left\{\begin{array}{r}
{0.7 x-0.2 y=-1.6} \\
{0.2 x-y=-1.4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 62

Solve each system of equations.
$$
\left\{\begin{array}{c}
{-0.7 x+0.6 y=1.3} \\
{0.5 x-0.3 y=-0.8}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 63

Solve each system of equations.
$$
\left\{\begin{array}{l}
{4 x-1.5 y=10.2} \\
{2 x+7.8 y=-25.68}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 64

Solve each system of equations.
$$
\left\{\begin{array}{r}
{x-3 y=-5.3} \\
{6.3 x+6 y=3.96}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 65

Determine whether the given replacement values make equation true or false. See Section 1.3.
$$
3 x-4 y+2 z=5 ; x=1, y=2, \text { and } z=5
$$

Christopher Stanley

Christopher Stanley

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Problem 66

Determine whether the given replacement values make equation true or false. See Section 1.3.
$$
x+2 y-z=7 ; x=2, y=-3, \text { and } z=3
$$

Christopher Stanley

Christopher Stanley

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Problem 67

Determine whether the given replacement values make equation true or false. See Section 1.3.
$$
-x-5 y+3 z=15 ; x=0, y=-1, \text { and } z=5
$$

Christopher Stanley

Christopher Stanley

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Problem 68

Determine whether the given replacement values make equation true or false. See Section 1.3.
$$
-4 x+y-8 z=4 ; x=1, y=0, \text { and } z=-1
$$

Christopher Stanley

Christopher Stanley

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Problem 69

Add the equations. See Section 4.1.
$$
\begin{aligned}
&3 x+2 y-5 z=10\\
&-3 x+4 y+z=15
\end{aligned}
$$

Christopher Stanley

Christopher Stanley

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Problem 70

Add the equations. See Section 4.1.
$$
\begin{aligned}
x+4 y-5 z &=20 \\
2 x-4 y-2 z &=-17
\end{aligned}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 71

Add the equations. See Section 4.1.
$$
\begin{array}{l}
{10 x+5 y+6 z=14} \\
{-9 x+5 y-6 z=-12}
\end{array}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 72

Add the equations. See Section 4.1.
$$
\begin{array}{r}
{-9 x-8 y-z=31} \\
{9 x+4 y-z=12}
\end{array}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 73

Without graphing, determine whether each system has one solution, no solution, or an infinite number of solutions. See the second Concept Check in this section.
$$
\left\{\begin{array}{l}
{y=2 x-5} \\
{y=2 x+1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 74

Without graphing, determine whether each system has one solution, no solution, or an infinite number of solutions. See the second Concept Check in this section.
$$
\left\{\begin{array}{l}
{y=3 x-\frac{1}{2}} \\
{y=-2 x+\frac{1}{5}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 75

Without graphing, determine whether each system has one solution, no solution, or an infinite number of solutions. See the second Concept Check in this section.
$$
\left\{\begin{array}{r}
{x+y=3} \\
{5 x+5 y=15}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 76

Without graphing, determine whether each system has one solution, no solution, or an infinite number of solutions. See the second Concept Check in this section.
$$
\left\{\begin{array}{l}
{y=5 x-2} \\
{y=-\frac{1}{5} x-2}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

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Problem 77

Can a system consisting of two linear equations have exactly two solutions? Explain why or why not.

Christopher Stanley

Christopher Stanley

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Problem 78

Suppose the graph of the equations in a system of two equations in two variables consists of a circle and a line. Discuss the possible number of solutions for this system.

Christopher Stanley

Christopher Stanley

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Problem 79

The concept of supply and demand is used often in business. In general, as the unit price of a commodity increases, the demand for that commodity decreases. Also, as a commodity's unit price increases, the manufacturer normally increases the supply. The point where supply is equal to demand is called the equilibrium point. The following shows the graph of a demand equation and the graph of a supply equation for previously rented DVDs. The $x$ -axis represents the number of DVDs in thousands, and the $y$ -axis represents the cost of a DVD. Use this graph to answer Exercises 79 through 82. (GRAPH CANNOT COPY).
Find the number of DVDs and the price per DVD when supply equals demand.

Christopher Stanley

Christopher Stanley

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Problem 80

The concept of supply and demand is used often in business. In general, as the unit price of a commodity increases, the demand for that commodity decreases. Also, as a commodity's unit price increases, the manufacturer normally increases the supply. The point where supply is equal to demand is called the equilibrium point. The following shows the graph of a demand equation and the graph of a supply equation for previously rented DVDs. The $x$ -axis represents the number of DVDs in thousands, and the $y$ -axis represents the cost of a DVD. Use this graph to answer Exercises 79 through 82. (GRAPH CANNOT COPY).
When $x$ is between 3 and $4,$ is supply greater than demand or is demand greater than supply?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 81

The concept of supply and demand is used often in business. In general, as the unit price of a commodity increases, the demand for that commodity decreases. Also, as a commodity's unit price increases, the manufacturer normally increases the supply. The point where supply is equal to demand is called the equilibrium point. The following shows the graph of a demand equation and the graph of a supply equation for previously rented DVDs. The $x$ -axis represents the number of DVDs in thousands, and the $y$ -axis represents the cost of a DVD. Use this graph to answer Exercises 79 through 82. (GRAPH CANNOT COPY).
When $x$ is greater than $6,$ is supply greater than demand or is demand greater than supply?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 82

The concept of supply and demand is used often in business. In general, as the unit price of a commodity increases, the demand for that commodity decreases. Also, as a commodity's unit price increases, the manufacturer normally increases the supply. The point where supply is equal to demand is called the equilibrium point. The following shows the graph of a demand equation and the graph of a supply equation for previously rented DVDs. The $x$ -axis represents the number of DVDs in thousands, and the $y$ -axis represents the cost of a DVD. Use this graph to answer Exercises 79 through 82. (GRAPH CANNOT COPY).
For what $x$ -values are the $y$ -values corresponding to the supply equation greater than the $y$ -values corresponding to the demand equation?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 83

The revenue equation for a certain brand of toothpaste is $y=2.5 x$ where $x$ is the number of tubes of toothpaste sold and $y$ is the total income for selling $x$ tubes. The cost equation is $y=0.9 x+3000$ where $x$ is the number of tubes of toothpaste manufactured and $y$ is the cost of producing $x$ tubes. The following set of axes shows the graph of the cost and revenue equations. Use this graph for Exercises 83 through 88. (GRAPH CANNOT COPY).
Find the coordinates of the point of intersection, or break-even point, by solving the system
$$
\left\{\begin{array}{l}
{y=2.5 x} \\
{y=0.9 x+3000}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 84

The revenue equation for a certain brand of toothpaste is $y=2.5 x$ where $x$ is the number of tubes of toothpaste sold and $y$ is the total income for selling $x$ tubes. The cost equation is $y=0.9 x+3000$ where $x$ is the number of tubes of toothpaste manufactured and $y$ is the cost of producing $x$ tubes. The following set of axes shows the graph of the cost and revenue equations. Use this graph for Exercises 83 through 88. (GRAPH CANNOT COPY).
Explain the meaning of the $x$ -value of the point of intersection.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 85

The revenue equation for a certain brand of toothpaste is $y=2.5 x$ where $x$ is the number of tubes of toothpaste sold and $y$ is the total income for selling $x$ tubes. The cost equation is $y=0.9 x+3000$ where $x$ is the number of tubes of toothpaste manufactured and $y$ is the cost of producing $x$ tubes. The following set of axes shows the graph of the cost and revenue equations. Use this graph for Exercises 83 through 88. (GRAPH CANNOT COPY).
If the company sells 2000 tubes of toothpaste, does the company make money or lose money?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 86

The revenue equation for a certain brand of toothpaste is $y=2.5 x$ where $x$ is the number of tubes of toothpaste sold and $y$ is the total income for selling $x$ tubes. The cost equation is $y=0.9 x+3000$ where $x$ is the number of tubes of toothpaste manufactured and $y$ is the cost of producing $x$ tubes. The following set of axes shows the graph of the cost and revenue equations. Use this graph for Exercises 83 through 88. (GRAPH CANNOT COPY).
If the company sells 1000 tubes of toothpaste, does the company make money or lose money?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 87

The revenue equation for a certain brand of toothpaste is $y=2.5 x$ where $x$ is the number of tubes of toothpaste sold and $y$ is the total income for selling $x$ tubes. The cost equation is $y=0.9 x+3000$ where $x$ is the number of tubes of toothpaste manufactured and $y$ is the cost of producing $x$ tubes. The following set of axes shows the graph of the cost and revenue equations. Use this graph for Exercises 83 through 88. (GRAPH CANNOT COPY).
For what $x$ -values will the company make a profit? (Hint: For what $x$ -values is the revenue graph "higher" than the cost graph?)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 88

The revenue equation for a certain brand of toothpaste is $y=2.5 x$ where $x$ is the number of tubes of toothpaste sold and $y$ is the total income for selling $x$ tubes. The cost equation is $y=0.9 x+3000$ where $x$ is the number of tubes of toothpaste manufactured and $y$ is the cost of producing $x$ tubes. The following set of axes shows the graph of the cost and revenue equations. Use this graph for Exercises 83 through 88. (GRAPH CANNOT COPY).
For what $x$ -values will the company lose money? (Hint: For what $x$ -values is the revenue graph "lower" than the cost graph?)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 89

Write a system of two linear equations in $x$ and $y$ that has the ordered pair solution $(2,5)$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 90

Which method would you use to solve the system?
$$
\left\{\begin{array}{l}
{5 x-2 y=6} \\
{2 x+3 y=5}
\end{array}\right.
$$
Explain your choice.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 91

The amount $y$ of bottled water consumed per person in the United States (in gallons) in the year $x$ can be modeled by the linear equation $y=1.47 x+9.26 .$ The amount $y$ of carbonated diet soft drinks consumed per person in the United States (in gallons) in the year $x$ can be modeled by the linear equation $y=0.13 x+13.55 .$ In both models, $x=0$ represents the year $1995 .$ (Source: Based on data from the Economic Research Service, U.S. Department of Agriculture)
a. What does the slope of each equation tell you about the patterns of bottled water and carbonated diet soft drink consumption in the United States?
b. Solve this system of equations. (Round your final results to the nearest whole numbers.)
c. Explain the meaning of your answer to part (b).

Foster Wisusik

Numerade Educator

Problem 92

The amount of U.S. federal government income $y$ (in billions of dollars) for fiscal year $x,$ from 2006 through $2009(x=0$ represents $2006),$ can be modeled by the linear equation $y=-95 x+2406 .$ The amount of U.S. federal government expenditures $y$ (in billions of dollars) for the same period can be modeled by the linear equation $y=285 x+2655$ (Source: Based on data from Financial Management Service, U.S. Department of the Treasury, $2006-2009$ )
a. What does the slope of each equation tell you about the patterns of U.S. federal government income and expenditures?
b. Solve this system of equations. (Round your final results to the nearest whole numbers.)
c. Did expenses ever equal income during the period from 2006 through $2009 ?$

Lewis Groves

Numerade Educator

Problem 93

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{l}
{\frac{1}{x}+y=12} \\
{\frac{3}{x}-y=4}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 94

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{c}
{x+\frac{2}{y}=7} \\
{3 x+\frac{3}{y}=6}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 95

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{l}
{\frac{1}{x}+\frac{1}{y}=5} \\
{\frac{1}{x}-\frac{1}{y}=1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 96

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{l}
{\frac{2}{x}+\frac{3}{y}=5} \\
{\frac{5}{x}-\frac{3}{y}=2}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 97

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{l}
{\frac{2}{x}-\frac{4}{y}=5} \\
{\frac{1}{x}-\frac{2}{y}=\frac{3}{2}}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 98

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{l}
{\frac{2}{x}+\frac{3}{y}=-1} \\
{\frac{3}{x}-\frac{2}{y}=18}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 99

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{l}
{\frac{3}{x}-\frac{2}{y}=-18} \\
{\frac{2}{x}+\frac{3}{y}=1}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 100

Solve each system. To do so, you may want to let $a=\frac{1}{x}$ (if $x$ is in the denominator) and let $b=\frac{1}{y}$ (if $y$ is in the denominator.)
$$
\left\{\begin{array}{r}
{\frac{5}{x}+\frac{7}{y}=1} \\
{-\frac{10}{x}-\frac{14}{y}=0}
\end{array}\right.
$$

Christopher Stanley

Christopher Stanley

Numerade Educator