Introductory and Intermediate Algebra for College Students

Robert Blitzer

Chapter 3

Systems of Linear Equations - all with Video Answers

Educators

AG

Section 1

Systems of Linear Equations in Two Variables

Problem 1

In Exercises $1-6,$ determine whether the given ordered pair is a solution of the system.
$(7,-5)$
$$
\left\{\begin{array}{l}{x-y=12} \\ {x+y=2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 2

Determine whether the given ordered pair is a solution of the system.
$(-3,1)$
$$
\left\{\begin{array}{c}{x-y=-4} \\ {2 x+10 y=4}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 3

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

AG

Ankit Gupta

Numerade Educator

Problem 4

Determine whether the given ordered pair is a solution of the system.
(4, 2)
$$
\left\{\begin{array}{l}{2 x-5 y=-2} \\ {3 x+4 y=18}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 5

Determine whether the given ordered pair is a solution of the system.
(5, -3)
$$
\left\{\begin{array}{l}{y=2 x-13} \\ {4 x+9 y=-7}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 6

Determine whether the given ordered pair is a solution of the system.
(-3, -4)
$$
\left\{\begin{array}{l}{y=3 x+5} \\ {5 x-2 y=-7}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 7

In Exercises $7-24,$ solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x+y=4} \\ {x-y=2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 8

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x+y=6} \\ {x-y=-4}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 9

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+y=4} \\ {y=4 x+1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 10

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x+2 y=4} \\ {y=-2 x-1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 11

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x-2 y=6} \\ {x-4 y=-8}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 12

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{4 x+y=4} \\ {3 x-y=3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 13

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+3 y=6} \\ {4 x=-6 y+12}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 14

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x-3 y=6} \\ {2 x=2 y+4}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 15

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{y=2 x-2} \\ {y=-5 x+5}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 16

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{y=-x+1} \\ {y=3 x+5}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 17

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x-y=4} \\ {6 x-2 y=4}\end{array}\right.
$$

Check back soon!

Problem 18

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x-y=-4} \\ {4 x-2 y=6}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 19

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+y=4} \\ {4 x+3 y=10}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 20

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{4 x-y=9} \\ {x-3 y=16}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 21

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-y=2} \\ {y=1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 22

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x+2 y=1} \\ {x=3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 23

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x+y=3} \\ {6 x+2 y=12}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 24

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x-3 y=6} \\ {4 x-6 y=24}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 25

In Exercises $25-42,$ solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x+y=6} \\ {y=2 x}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 26

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x+y=10} \\ {y=4 x}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 27

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+3 y=9} \\ {x=y+2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 28

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x-4 y=18} \\ {y=1-2 x}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 29

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{y=-3 x+7} \\ {5 x-2 y=8}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 30

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x=3 y+8} \\ {2 x-y=6}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 31

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{4 x+y=5} \\ {2 x-3 y=13}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 32

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-3 y=3} \\ {3 x+5 y=-19}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 33

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-2 y=4} \\ {2 x-4 y=5}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 34

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-3 y=6} \\ {2 x-6 y=5}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 35

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+5 y=-4} \\ {3 x-y=11}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 36

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+5 y=1} \\ {-x+6 y=8}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 37

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2(x-1)-y=-3} \\ {y=2 x+3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 38

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{aligned} x+y-1 &=2(y-x) \\ y &=3 x-1 \end{aligned}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 39

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{\frac{x}{4}-\frac{y}{4}=-1} \\ {x+4 y=-9}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 40

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{\frac{x}{6}-\frac{y}{2}=\frac{1}{3}} \\ {x+2 y=-3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 41

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{y=\frac{2}{5} x-2} \\ {2 x-5 y=10}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 42

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{y=\frac{1}{3} x+4} \\ {3 y=x+12}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 43

In Exercises $43-58,$ solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{x+y=7} \\ {x-y=3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 44

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{2 x+y=3} \\ {x-y=3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 45

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{12 x+3 y=15} \\ {2 x-3 y=13}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 46

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{4 x+2 y=12} \\ {3 x-2 y=16}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 47

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x+3 y=2} \\ {4 x+5 y=1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 48

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{aligned} x+2 y &=-1 \\ 2 x-y &=3 \end{aligned}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 49

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{6 x-y=-5} \\ {4 x-2 y=6}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 50

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-2 y=5} \\ {5 x-y=-2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 51

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x-5 y=11} \\ {2 x-6 y=2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 52

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{4 x-3 y=12} \\ {3 x-4 y=2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 53

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x-5 y=13} \\ {5 x+3 y=17}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 54

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{4 x+5 y=-9} \\ {6 x-3 y=-3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 55

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x+6 y=8} \\ {3 x+9 y=12}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 56

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-3 y=-6} \\ {3 x-9 y=9}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 57

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{2 x-3 y=4} \\ {4 x+5 y=3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 58

Solve each system by the addition method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{4 x-3 y=8} \\ {2 x-5 y=-14}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 59

In Exercises $59-82,$ solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{3 x-7 y=1} \\ {2 x-3 y=-1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 60

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{2 x-3 y=2} \\ {5 x+4 y=51}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 61

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{x=y+4} \\ {3 x+7 y=-18}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 62

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{y=3 x+5} \\ {5 x-2 y=-7}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 63

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{9 x+\frac{4 y}{3}=5} \\ {4 x-\frac{y}{3}=5}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 64

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{\frac{x}{6}-\frac{y}{5}=-4} \\ {\frac{x}{4}-\frac{y}{6}=-2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 65

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{\frac{1}{4} x-\frac{1}{9} y=\frac{2}{3}} \\ {\frac{1}{2} x-\frac{1}{3} y=1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 66

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{\frac{1}{16} x-\frac{3}{4} y=-1} \\ {\frac{3}{4} x+\frac{5}{2} y=11}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 67

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{x=3 y-1} \\ {2 x-6 y=-2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 68

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{x=4 y-1} \\ {2 x-8 y=-2}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 69

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{y=2 x+1} \\ {y=2 x-3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 70

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{y=2 x+4} \\ {y=2 x-1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 71

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{0.4 x+0.3 y=2.3} \\ {0.2 x-0.5 y=0.5}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 72

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{0.2 x-y=-1.4} \\ {0.7 x-0.2 y=-1.6}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 73

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{5 x-40=6 y} \\ {2 y=8-3 x}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 74

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{4 x-24=3 y} \\ {9 y=3 x-1}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 75

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{3(x+y)=6} \\ {3(x-y)=-36}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 76

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{4(x-y)=-12} \\ {4(x+y)=-20}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 77

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{3(x-3)-2 y=0} \\ {2(x-y)=-x-3}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 78

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{c}{5 x+2 y=-5} \\ {4(x+y)=6(2-x)}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 79

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{x+2 y-3=0} \\ {12=8 y+4 x}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 80

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{array}{l}{2 x-y-5=0} \\ {10=4 x-2 y}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 81

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{aligned} 3 x+4 y &=0 \\ 7 x &=3 y \end{aligned}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 82

Solve each system by the method of your choice. Identify inconsistent systems and systems with dependent equations, using set notation to express solution sets.
$$
\left\{\begin{aligned} 5 x+8 y &=20 \\ 4 y &=-5 x \end{aligned}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 83

In Exercises $83-84,$ solve each system by the method of your choice.
$$
\left\{\begin{array}{l}{\frac{x+2}{2}-\frac{y+4}{3}=3} \\ {\frac{x+y}{5}=\frac{x-y}{2}-\frac{5}{2}}\end{array}\right.
$$

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 84

Solve each system by the method of your choice.
$$
\left\{\begin{array}{l}{\frac{x-y}{3}=\frac{x+y}{2}-\frac{1}{2}} \\ {\frac{x+2}{2}-4=\frac{y+4}{3}}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 85

In Exercises $85-86,$ solve each system for $x$ and $y,$ expressing either value in terms of a or $b,$ if necessary. Assume that $a \neq 0$ and $b \neq 0$.
$$
\left\{\begin{array}{c}{5 a x+4 y=17} \\ {a x+7 y=22}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 86

Solve each system for $x$ and $y,$ expressing either value in terms of a or $b,$ if necessary. Assume that $a \neq 0$ and $b \neq 0$.
$$
\left\{\begin{array}{l}{4 a x+b y=3} \\ {6 a x+5 b y=8}\end{array}\right.
$$

Erika Bustos

Numerade Educator

Problem 87

For the linear function $f(x)=m x+b, f(-2)=11$ and $f(3)=-9 .$ Find $m$ and $b .$

AG

Ankit Gupta

Numerade Educator

Problem 88

For the linear function $f(x)=m x+b, f(-3)=23$ and $f(2)=-7 .$ Find $m$ and $b$

AG

Ankit Gupta

Numerade Educator

Problem 89

Use the graphs of the linear functions to solve Exercises 89–90.
$$
\begin{array}{l}{\text { Write the linear system whose solution set is }\{(6,2)\} .} \\ {\text { Express each equation in the system in slope-intercept form. }}\end{array}
$$
(Graph cannot copy)

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 90

Write the linear system whose solution set is $\varnothing .$ Express each equation in the system in slope-intercept form.
(Graph cannot copy)

AG

Ankit Gupta

Numerade Educator

Problem 91

We opened the section with a bar graph that showed fewer U.S. adults are getting married. (See Figure 3.1 on page $178 .)$ The data can be modeled by the following system of linear equations:
$$
\left\{\begin{aligned}-3 x+10 y &=160 \\ x+2 y &=142 \end{aligned}\right.
$$
$$
\begin{array}{l}{\text { a. Use these models to determine the year, rounded to }} \\ {\text { the nearest year, when the percentage of never-married }} \\ {\text { adults will be the same as the percentage of married }} \\ {\text { adults. For that year, approximately what percentage of }} \\ {\text { Americans, rounded to the nearest percent, will belong }} \\ {\text { to each group? }} \\ {\text { b. How is your approximate solution from part (a) shown }} \\ {\text { by the following graphs? }}\end{array}
$$
(Graph cannot copy)

AG

Ankit Gupta

Numerade Educator

Problem 92

The graph shows that from 2000 through 2006 , Americans unplugged land lines and switched to cellphones. a. Use the graphs to estimate the point of intersection. In what year was the number of cellphone and land-line customers the same? How many millions of customers were there for each? b. The function $4.3 x+y=198$ models the number of land-line customers, in millions, $x$ years after 2000 . The function $y=19.8 x+98$ models the number of
cellphone customers, in millions, $x$ years after 2000 . Use these models to determine the year, rounded to the nearest year, when the number of cellphone and land-line customers was the same. According to the models, how many millions of customers, rounding to the nearest ten million, were there for each? c. How well do the models in part (b) describe the point of intersection of the graphs that you estimated in part (a)?

AG

Ankit Gupta

Numerade Educator

Problem 93

Although Social Security is a problem, some projections indicate that there's a much bigger time bomb ticking in the federal budget, and that's Medicare. In 2000 , the cost of Social Security was $5.48 \%$ of the gross domestic 2000 , the cost of Medicare was $1.84 \%$ of the gross domestic product, increasing by $0.17 \%$ of the GDP per year. (Source: Congressional Budget Office) a. Write a function that models the cost of Social Security as a percentage of the GDP $x$ years after 2000 . b. Write a function that models the cost of Medicare as a percentage of the GDP $x$ years after 2000 .
c. In which year will the cost of Medicare and Social Security be the same? For that year, what will be the cost of each program as a percentage of the GDP? Which program will have the greater cost after that year?

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 94

The Rise and Fall of D and E The graph indicates that vitamin D sales increased from 2001 to 2009, a period during which several studies linked it to various health benefits. News that vitamin E might increase the risk of illness and death resulted in decreasing sales for the same period. In $2001,$ vitamin D sales were 40 dollar million. For the period shown, sales increased by an average of 48 dollar million per year. In 2001 , vitamin E sales were 771 dollar million, decreasing by an average of 54 dollar million per year. a. Write a function that models the sale of vitamin $D,$ in millions of dollars, $x$ years after $2001 .$ b. Write a function that models the sale of vitamin $\mathrm{E},$ in millions of dollars, $x$ years after $2001 .$ c. In which year, to the nearest whole year, were sales of vitamin D and vitamin E the same?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 95

The bar graph shows the percentage of Americans who used cigarettes, by ethnicity, in 1985 and $2005 .$ For each of the groups shown, cigarette use has been linearly decreasing. Use this information to solve Exercises $95-96 .$ In this exercise, let $x$ represent the number of years after 1985 and let $y$ represent the percentage of Americans in one of the groups shown who used cigarettes. a. Use the data points $(0,38)$ and $(20,27.3)$ to find the slope-intercept equation of the line that models the percentage of African Americans who used cigarettes, $y$ $x$ years after $1985 .$ Round the value of the slope $m$ to two decimal places. b. Use the data points $(0,40)$ and $(20,24.2)$ to find the slope-intercept equation of the line that models the percentage of Hispanics who used cigarettes, $y, x$ years after $1985 .$ c. Use the models from parts (a) and (b) to find the year during which cigarette use was the same for African Americans and Hispanics. What percentage of each group used cigarettes during that year?

Jodi Folley

Numerade Educator

Problem 96

In this exercise, let $x$ represent the number of years after 1985 and let $y$ represent the percentage of Americans in one of the groups shown who used cigarettes. a. Use the data points $(0,38.9)$ and $(20,27.3)$ to find the slope-intercept equation of the line that models the percentage of whites who used cigarettes, $y, x$ years after $1985 .$ b. Use the data points $(0,40)$ and $(20,24.2)$ to find the slope-intercept equation of the line that models the percentage of Hispanics who used cigarettes, $y, x$ years after $1985 .$ c. Use the models from parts (a) and (b) to find the year, to
the nearest whole year, during which cigarette use was the same for whites and Hispanics. What percentage of each group, to the nearest percent, used cigarettes during that year?

Ethan Somes

Numerade Educator

Problem 97

An important application of systems of equations arises in connection with supply and demand. As the price of a product increases, the demand for that product decreases. However, at higher prices, suppliers are willing to produce greater quantities of the product. Exercises $97-98$ involve supply and demand. A chain of electronics stores sells hand-held color televisions. The weekly demand and supply models are given as follows:
$$
N=-5 p+750
$$
$$
N=2.5 p.
$$
a. How many hand-held color televisions can be sold and
supplied at $\$ 120$ per television?
b. Find the price at which supply and demand are equal.
At this price, how many televisions can be supplied and
sold each week?

Christy Galilei

Christy Galilei

Numerade Educator

Problem 98

At a price of $p$ dollars per ticket, the number of tickets to a rock concert that can be sold is given by the demand model $N=-25 p+7800$. At a price of $p$ dollars per ticket, the number of tickets that the concert's promoters are willing to make available is given by the supply model $N=5 p+6000$. a. How many tickets can be sold and supplied for $\$ 50$ per ticket? b. Find the ticket price at which supply and demand are equal. At this price, how many tickets will be supplied and sold?

Bobby Barnes

University of North Texas

Problem 99

What is a system of linear equations? Provide an example with your description.

John Irizar

Numerade Educator

Problem 100

What is a solution of a system of linear equations?

John Irizar

Numerade Educator

Problem 101

Explain how to determine if an ordered pair is a solution of a system of linear equations.

John Irizar

Numerade Educator

Problem 102

Explain how to solve a system of linear equations by graphing.

John Irizar

Numerade Educator

Problem 103

Explain how to solve a system of equations using the substitution method. Use $y=3-3 x$ and $3 x+4 y=6$ to illustrate your explanation.

Jake Zanazzi

Numerade Educator

Problem 104

Explain how to solve a system of equations using the addition method. Use $5 x+8 y=-1$ and $3 x+y=7$ to illustrate your explanation.

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 105

When is it easier to use the addition method rather than the substitution method to solve a system of equations?

AG

Ankit Gupta

Numerade Educator

Problem 106

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

AG

Ankit Gupta

Numerade Educator

Problem 107

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

AG

Ankit Gupta

Numerade Educator

Problem 108

Verify your solutions to any five exercises from Exercises $7-24$ by using a graphing utility to graph the two equations in the system in the same viewing rectangle. Then use the intersection feature to display the solution.

AG

Ankit Gupta

Numerade Educator

Problem 109

Make Sense? In Exercises $109-112,$ determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
$$
\begin{array}{l}{\text { Even if a linear system has a solution set involving }} \\ {\text { fractions, such as }\left\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\}, \text { I can use graphs to }} \\ {\text { determine if the solution set is reasonable. }}\end{array}
$$

AG

Ankit Gupta

Numerade Educator

Problem 110

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. If I add the equations on the right and solve the resulting equation for $x,$ I will obtain the $x$ -coordinate of the intersection point of the lines represented by the equations on the left.
$$
\left\{\begin{array}{l}{4 x-6 y=1 \longrightarrow} \\ {3 x+5 y=-8 \longrightarrow}\end{array}\left\{\begin{array}{l}{20 x-30 y=5} \\ {18 x+30 y=-8}\end{array}\right.\right.
$$

Vinnu M

Numerade Educator

Problem 111

In the previous chapter, we developed models for life expectancy, y, for U.S. men and women born $x$ years after 1960 :
$$
\left\{\begin{array}{l}{y=0.22 x+65.7} \\ {y=0.17 x+72.9}\end{array}\right.
$$
The system indicates that life expectancy for men is increasing at a faster rate than for women, so if these trends continue, life expectancies for men and women will be the same for some future birth year.

Cinsy Krehbiel

Numerade Educator

Problem 112

Here are two models that describe winning times for the Olympic 400 -meter run, $y,$ in seconds, $x$ years after 1968 :
$$
\left\{\begin{array}{l}{y=-0.02433 x+44.43} \\ {y=-0.08883 x+50.86}\end{array}\right.
$$
The system indicates that winning times have been decreasing more rapidly for women than for men, so if these trends continue, there will be a year when winning times for men and women are the same.

John Irizar

Numerade Educator

Problem 113

In Exercises $113-116$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$
\begin{array}{l}{\text { The addition method cannot be used to eliminate }} \\ {\text { either variable in a system of two equations in two }} \\ {\text { variables. }}\end{array}
$$

Matt Gibson

Numerade Educator

Problem 114

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The solution set of the system
$$
\left\{\begin{array}{c}{5 x-y=1} \\ {10 x-2 y=2}\end{array}\right.
$$
$$
\text { is }\{(2,9)\}
$$

Alexia Hamilton

Alexia Hamilton

Numerade Educator

Problem 115

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The solution set of the system A system of linear equations can have a solution set consisting of precisely two ordered pairs.

AG

Ankit Gupta

Numerade Educator

Problem 116

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The solution set of the system.
$$
\left\{\begin{array}{l}{y=4 x-3} \\ {y=4 x+5}\end{array}\right.
$$
is the empty set.

Nicole Hoffman

Numerade Educator

Problem 117

Determine $a$ and $b$ so that $(2,1)$ is a solution of this system:
$$
\left\{\begin{array}{l}{a x-b y=4} \\ {b x+a y+7}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 118

Write a system of equations having $\{(-2,7)\}$ as a solution set. (More than one system is possible.)

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 119

Solve the system for $x$ and $y$ in terms of $a_{1}, b_{1}, c_{1}, a_{2}, b_{2},$ and $c_{2}:$
$$
\left\{\begin{array}{l}{a_{1} x+b_{1} y=c_{1}} \\ {a_{2} x+b_{2} y=c_{2}}\end{array}\right.
$$

AG

Ankit Gupta

Numerade Educator

Problem 120

Review Exercises
Solve: $6 x=10+5(3 x-4) .$ (Section $1.4,$ Example 3 )

AG

Ankit Gupta

Numerade Educator

Problem 121

Simplify: $\left.\left(4 x^{2} y^{4}\right)^{2}\left(-2 x^{5} y^{0}\right)^{3} \text { . (Section } 1.6, \text { Example } 9\right)$

AG

Ankit Gupta

Numerade Educator

Problem 122

If $\left.f(x)=x^{2}-3 x+7, \text { find } f(-1) \text { . (Section } 2.1, \text { Example } 3\right)$

AG

Ankit Gupta

Numerade Educator

Problem 123

Exercises $123-125$ will help you prepare for the material covered in the next section. The formula $I=P r$ is used to find the simple interest, $I$, earned for one year when the principal, $P,$ is invested at an annual interest rate, $r$. Write an expression for the total interest earned on a principal of $x$ dollars at a rate of $15 \%(r=0.15)$ and a principal of $y$ dollars at a rate of $7 \%(r=0.07)$

Amrita Bhasin

Numerade Educator

Problem 124

A chemist working on a flu vaccine needs to obtain 50 milliliters of a $30 \%$ sodium-iodine solution. How many milliliters of sodium-iodine are needed in the solution?

David Collins

Numerade Educator

Problem 125

A company that manufactures running shoes sells them at $\$ 80 dollar per pair. Write an expression for the revenue that is generated by selling $x$ pairs of shoes.

Nick Johnson

Numerade Educator