Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 4

Systems of Equations in Two Variables

Educators


Problem 1

following statements as either true or false.
Solutions of systems of equations in two variables are ordered pairs.

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Problem 2

Classify each of the following statements as either true or false.
Every system of equations has at least one solution.

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Problem 3

Classify each of the following statements as either true or false.
It is possible for a system of equations to have an infinite number of solutions.

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Problem 4

Classify each of the following statements as either true or false.
The graphs of the equations in a system of two equations may coincide.

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Problem 5

Classify each of the following statements as either true or false.
The graphs of the equations in a system of two equations could be parallel lines.

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Problem 6

Classify each of the following statements as either true or false.
Any system of equations that has at most one solution is said to be consistent.

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Problem 7

Classify each of the following statements as either true or false.
Any system of equations that has more than one solution is said to be inconsistent.

Cory K.
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Problem 8

Classify each of the following statements as either true or false.
The equations $x+y=5$ and $2(x+y)=2(5)$ are dependent.

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Problem 9

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(1,2) ; & 4 x-y=2 \\
10 x-3 y &=4
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 10

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(4,0) ; & 2 x+7 y=8 \\
x-9 y &=4
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 11

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(-5,1) ; & x+5 y=0 \\
y=& 2 x+9
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 12

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(-1,-2) ; & x+3 y=-7 \\
3 x-2 y=& 12
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 13

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(0,-5) ; & x-y=5 \\
y=& 3 x-5
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 14

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(5,2) ; & a+b=7 \\
2 a-8 &=b
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 15

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(3,1) ; & 3 x+4 y=13 \\
& 6 x+8 y=26
\end{aligned}
$$

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Numerade Educator

Problem 16

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables.
$$
\begin{aligned}
(4,-2) ; &-3 x-2 y=-8 \\
8 &=3 x+2 y
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 17

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&x-y=3\\
&x+y=5
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 18

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{array}{l}
{x+y=4} \\
{x-y=2}
\end{array}
$$

Cory K.
Numerade Educator

Problem 19

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{array}{r}
{3 x+y=5} \\
{x-2 y=4}
\end{array}
$$

Cory K.
Numerade Educator

Problem 20

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&2 x-y=4\\
&5 x-y=13
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 21

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&4 y=x+8\\
&3 x-2 y=6
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 22

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{array}{r}
{4 x-y=9} \\
{x-3 y=16}
\end{array}
$$

Cory K.
Numerade Educator

Problem 23

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
x &=y-1 \\
2 x &=3 y
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 24

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&a=1+b\\
&b=5-2 a
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 25

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&x=-3\\
&y=2
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 26

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{array}{l}
{x=4} \\
{y=-5}
\end{array}
$$

Cory K.
Numerade Educator

Problem 27

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&t+2 s=-1\\
&s=t+10
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 28

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{array}{r}
{2 b+a=11} \\
{a-b=5}
\end{array}
$$

Cory K.
Numerade Educator

Problem 29

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{array}{r}
{2 b+a=11} \\
{a-b=5}
\end{array}
$$

Cory K.
Numerade Educator

Problem 30

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y=-\frac{1}{3} x-1\\
&4 x-3 y=18
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 31

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y=-\frac{1}{4} x+1\\
&2 y=x-4
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 32

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&6 x-2 y=2\\
&9 x-3 y=1
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 33

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y-x=5\\
&2 x-2 y=10
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 34

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y=-x-1\\
&4 x-3 y=24
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 35

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y=3-x\\
&2 x+2 y=6
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 36

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&2 x-3 y=6\\
&3 y-2 x=-6
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 37

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y=-5.43 x+10.89\\
&y=6.29 x-7.04
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 38

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&y=123.52 x+89.32\\
&y=-89.22 x+33.76
\end{aligned}
$$

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Problem 39

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&2.6 x-1.1 y=4\\
&1.32 y=3.12 x-5.04
\end{aligned}
$$

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Problem 40

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&2.18 x+7.81 y=13.78\\
&5.79 x-3.45 y=8.94
\end{aligned}
$$

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Problem 41

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&0.2 x-y=17.5\\
&2 y-10.6 x=30
\end{aligned}
$$

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Problem 42

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
$$
\begin{aligned}
&1.9 x=4.8 y+1.7\\
&12.92 x+23.8=32.64 y
\end{aligned}
$$

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Problem 43

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
For the systems in the odd-numbered exercises $17-41$ which are consistent?

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Problem 44

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
For the systems in the even-numbered exercises $18-42$ which are consistent?

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Problem 45

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
For the systems in the odd-numbered exercises $17-41$ which contain dependent equations?

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Problem 46

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
For the systems in the even-numbered exercises $18-42$ which contain dependent equations?

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Problem 47

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
College Faculty. The number of part-time faculty in institutions of higher learning is growing rapidly. The table below lists the number of full-time faculty and the number of part-time faculty for various years. Use linear regression to fit a line to each set of data, and use those equations to predict the year in which the number of part-time faculty and the number of fulltime faculty will be the same.
(TABLE CANNOT COPY)

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Problem 48

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
Milk Cows. The number of milk cows in Vermont has decreased since $2004,$ while the number in Colorado has increased, as shown in the table below. Use linear regression to fit a line to each set of data, and use those equations to predict the year in which the number of milk cows in the two states will be the same.
(TABLE CANNOT COPY)

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Problem 49

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
Financial Advisers. Financial advisers are leaving major national firms and setting up their own businesses,as shown in the table below. Use linear regression to fit a line to each set of data, and use those equations to predict the year in which there will be the same number of independent financial advisers as there are those in major national firms.
(TABLE CANNOT COPY)

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Problem 50

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
Recycling. In the United States, the amount of waste being recovered is slowly catching up to the amount of waste being discarded, as shown in the table below. Use linear regression to fit a line to each set of data, and use those equations to predict the year in which the amount of waste recycled will equal the amount discarded.
(TABLE CANNOT COPY)

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Problem 51

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
Suppose that the equations in a system of two linear equations are dependent. Does it follow that the system is consistent? Why or why not?

Cory K.
Numerade Educator

Problem 52

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If $a$
system has no solution, state this. Where appropriate, round to the nearest hundredth.
Why is slope-intercept form especially useful when solving systems of equations by graphing?

Cory K.
Numerade Educator

Problem 53

To prepare for Section $4.2,$ review solving equations and formulas (Sections 2.2 and 2.3 ).
Solve. [ 2.2]
$$
2(4 x-3)-7 x=9
$$

Cory K.
Numerade Educator

Problem 54

To prepare for Section $4.2,$ review solving equations and formulas (Sections 2.2 and 2.3 ).
Solve. [ 2.2]
$$
6 y-3(5-2 y)=4
$$

Cory K.
Numerade Educator

Problem 55

To prepare for Section $4.2,$ review solving equations and formulas (Sections 2.2 and 2.3 ).
Solve. [ 2.2]
$$
4 x-5 x=8 x-9+11 x
$$

Cory K.
Numerade Educator

Problem 56

To prepare for Section $4.2,$ review solving equations and formulas (Sections 2.2 and 2.3 ).
Solve. [ 2.2]
$$
8 x-2(5-x)=7 x+3
$$

Cory K.
Numerade Educator

Problem 57

Solve. [ 2.3]
$$
3 x+4 y=7, \text { for } y
$$

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Numerade Educator

Problem 58

Solve. [ 2.3]
$$
2 x-5 y=9, \text { for } y
$$

Cory K.
Numerade Educator

Problem 59

Consider the graph below showing the U.S. market share for various advertising media.
(GRAPH CANNOT COPY)
At what point in time could it have been said that no medium was third in market share? Explain.

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Problem 60

Consider the graph below showing the U.S. market share for various advertising media.
(GRAPH CANNOT COPY)
Will the Internet advertising market share ever exceed that of radio? TV? newspapers? If so, when? Explain your answers.

Cory K.
Numerade Educator

Problem 61

For each of the following conditions, write a system of equations.
A) $(5,1)$ is a solution.
B) There is no solution.
C) There is an infinite number of solutions.

Cory K.
Numerade Educator

Problem 62

A system of linear equations has $(1,-1)$ and $(-2,3)$ as solutions. Determine:
A) a third point that is a solution, and
B) how many solutions there are.

Cory K.
Numerade Educator

Problem 63

The solution of the following system is $(4,-5) .$ Find $A$ and $B$
$$
\begin{array}{l}
{A x-6 y=13} \\
{x-B y=-8}
\end{array}
$$

Cory K.
Numerade Educator

Problem 64

Solve graphically.
$$
\begin{aligned}
y &=|x| \\
x+4 y &=15
\end{aligned}
$$

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Numerade Educator

Problem 65

Solve graphically.
$$
\begin{aligned}
x-y &=0 \\
y &=x^{2}
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 66

Match each system with the appropriate graph from the selections given.
(IMAGE CANNOT COPY)$$
\begin{aligned}
&x=4 y\\
&3 x-5 y=7
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 67

Match each system with the appropriate graph from the selections given.
(GRAPH CANNOT COPY)
$$
\begin{aligned}
&2 x-8=4 y\\
&x-2 y=4
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 68

Match each system with the appropriate graph from the selections given.
(GRAPH CANNOT COPY)
$$
\begin{aligned}
&8 x+5 y=20\\
&4 x-3 y=6
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 69

Match each system with the appropriate graph from the selections given.
(GRAPH CANNOT COPY)
$$
\begin{aligned}
&x=3 y-4\\
&2 x+1=6 y
\end{aligned}
$$

Cory K.
Numerade Educator

Problem 70

Copying Costs. Aaron occasionally goes to an office store with small copying jobs. He can purchase a "copy card" for $\$ 20$ that will entitle him to 500 copies, or he can simply pay 6 per page.
A) Create cost equations for each method of paying for a number (up to 500 ) of copies.
B) Graph both cost equations on the same set of axes.
C) Use the graph to determine how many copies Aaron must make if the card is to be more economical.

Cory K.
Numerade Educator

Problem 71

Computers. In $2004,$ about 46 million notebook PCs were shipped worldwide, and that number was growing at a rate of $22 \frac{2}{3}$ million per year. There were 140 million desktop PCs shipped worldwide in 2004 and that number was growing at a rate of 4 million per year. Source: iSuppli
A) Write two equations that can be used to predict $n$ the number of notebook $\mathrm{PCs}$ and desktop $\mathrm{PCs},$ in millions, shipped $t$ years after 2004
B) Use a graphing calculator to determine the year in which the numbers of notebook PCs and desktop PCs were the same.

Cory K.
Numerade Educator