Intermediate Algebra : Concepts and Applications

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

Chapter 3

Systems of Linear Equations and Problem Solving - all with Video Answers

Educators

AG

Section 1

Systems of Equations in Two Variables

Problem 1

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

Cory Kuzinski

Cory Kuzinski

Numerade Educator

Problem 2

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.

Cory Kuzinski

Numerade Educator

Problem 3

Classify each of the following statements as either true or false.
Every point of intersection of the graphs of the equations in a system corresponds to a solution of the system.

AG

Ankit Gupta

Numerade Educator

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.

Cory Kuzinski

Numerade Educator

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.

Cory Kuzinski

Numerade Educator

Problem 6

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.

Cory Kuzinski

Numerade Educator

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 Kuzinski

Numerade Educator

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.

Cory Kuzinski

Numerade Educator

Problem 9

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

AG

Ankit Gupta

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.
$$
\begin{array}{r}
{(4,0) ; 2 x+7 y=8} \\
{x-9 y=4}
\end{array}
$$

Cory Kuzinski

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 Kuzinski

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 Kuzinski

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 Kuzinski

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 Kuzinski

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.
$$
\begin{array}{r}
{(3,-1) ; 3 x-4 y=13} \\
{6 x-8 y=26}
\end{array}
$$

AG

Ankit Gupta

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 Kuzinski

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.
$$
\begin{aligned}
&x-y=1\\
&x+y=5
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&x+y=6\\
&x-y=4
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{array}{r}
{3 x+y=5} \\
{x-2 y=4}
\end{array}
$$

AG

Ankit Gupta

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.
$$
\begin{array}{l}
{2 x-y=4} \\
{5 x-y=13}
\end{array}
$$

AG

Ankit Gupta

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

AG

Ankit Gupta

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.
$$
\begin{aligned}
4 x-y &=9 \\
x-3 y &=16
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
x &=y-1 \\
2 x &=3 y
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&a=1+b\\
&b=5-2 a
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y=-1\\
&x=3
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y=2\\
&x=-4
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&t+2 s=-1\\
&s=t+10
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&b+2 a=2\\
&a=-3-b
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&2 b+a=11\\
&a-b=5
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y=-\frac{1}{3} x-1\\
&4 x-3 y=18
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y=-\frac{1}{4} x+1\\
&2 y=x-4
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&6 x-2 y=2\\
&9 x-3 y=1
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y-x=5\\
&2 x-2 y=10
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y=x+2\\
&3 y-2 x=4
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&y=3-x\\
&2 x+2 y=6
\end{aligned}
$$

AG

Ankit Gupta

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.
$$
\begin{aligned}
&2 x-3 y=6\\
&3 y-2 x=-6
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 37

For the systems in the odd-numbered exercises $17-35$ which are consistent?

AG

Ankit Gupta

Numerade Educator

Problem 38

For the systems in the even-numbered exercises $18-36$ which are consistent?

AG

Ankit Gupta

Numerade Educator

Problem 39

For the systems in the odd-numbered exercises $17-35$ which contain dependent equations?

AG

Ankit Gupta

Numerade Educator

Problem 40

For the systems in the even-numbered exercises $18-36$ which contain dependent equations?

AG

Ankit Gupta

Numerade Educator

Problem 41

Translate each problem situation to a system of equations. Do not attempt to solve, but save for later use.
The sum of two numbers is $10 .$ The first number is $\frac{2}{3}$ of the second number. What are the numbers?

AG

Ankit Gupta

Numerade Educator

Problem 42

Translate each problem situation to a system of equations. Do not attempt to solve, but save for later use. The sum of two numbers is 30 . The first number is twice the second number. What are the numbers?

AG

Ankit Gupta

Numerade Educator

Problem 43

There were 1223 endangered plant and animal species in the United States. There were 243 more endangered plant species than animal species. How many plant species and how many animal species were considered endangered in $2016 ?$

Mukesh Devi

Numerade Educator

Problem 44

A nontoxic wood furniture polish can be made by mixing mineral (or olive) oil with vinegar. To make a 16 -oz batch for a squirt bottle, Jazmun uses an amount of mineral oil that is 4 oz more than twice the amount of vinegar. How much of each ingredient is required?

AG

Ankit Gupta

Numerade Educator

Problem 45

Two angles are supplementary." One angle is $3^{\circ}$ less than twice the other. Find the measures of the angles.
(GRAPH CANT COPY)

AG

Ankit Gupta

Numerade Educator

Problem 46

Two angles are complementary.' The sum of the measures of the first angle and half the second angle is $64^{\circ} .$ Find the measures of the angles.
(GRAPH CANT COPY)

Alayna Handiak

Numerade Educator

Problem 47

Wilt Chamberlain once scored 100 points, setting a record for points scored in an NBA game. Chamberlain took only two-point shots and (one-point) foul shots and made a total of 64 shots. How many shots of each type did he make?

AG

Ankit Gupta

Numerade Educator

Problem 48

The Fenton College Cougars made 40 field goals in a recent basketball game, some 2 -pointers and the rest 3 -pointers. Altogether, the 40 baskets counted for 89 points. How many of each type of field goal was made?

AG

Ankit Gupta

Numerade Educator

Problem 49

Anna purchased 32 strings for her autoharp. Wrapped strings cost $\$ 4.49$ each and unwrapped strings cost $\$ 2.99$ each. If she paid a total of $\$ 107.68$ for the strings, how many of each type did she buy?

AG

Ankit Gupta

Numerade Educator

Problem 50

Cool Treats sold 60 ice cream cones. single-dip cones sold for $\$ 2.50$ each and double-dip cones for $\$ 4.15$ each. In all, $\$ 179.70$ was taken in for the cones. How many of each size cone were sold?

Ashley High

Numerade Educator

Problem 51

Unraveled Knitters is an online group that knits items for nursing homes and shelters. For a recent campaign, they spent a total of 1072 hr knitting hats and scarves. Each hat takes 8 hr to knit and each scarf takes 12 hr to knit. If they donated 110 items, how many of each did they knit?

AG

Ankit Gupta

Numerade Educator

Problem 52

The Buck Creek Fire Department served 250 dinners. A child's plate cost $\$ 5.50$ and an adult's plate cost $\$ 9.00 .$ A total of $\$ 1935$ was collected. How many of each type of plate were served?

AG

Ankit Gupta

Numerade Educator

Problem 53

The perimeter of an NCAA men's lacrosse field is 340 yd. The length is 50 yd longer than the width. Find the dimensions
(IMAGE CANT COPY)

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Ankit Gupta

Numerade Educator

Problem 54

The perimeter of a standard tennis court used for doubles is $228 \mathrm{ft}$. The width is 42 ft less than the length. Find the dimensions.

AG

Ankit Gupta

Numerade Educator

Problem 55

Write a problem for a classmate to solve that requires writing a system of two equations. Devise the problem so that the solution is "The Bucks made 6 three-point baskets and 31 two-point baskets."

AG

Ankit Gupta

Numerade Educator

Problem 56

Write a problem for a classmate to solve that can be translated into a system of two equations. Devise the problem so that the solution is "In 2016 , Diana took five 3 -credit classes and two 4 -credit classes."

AG

Ankit Gupta

Numerade Educator

Problem 57

Simplify. Do not use negative exponents in the answer.
$$
-\frac{1}{2}-\frac{3}{10}
$$

AG

Ankit Gupta

Numerade Educator

Problem 58

Simplify. Do not use negative exponents in the answer.
$$
(0.05)(-1.2)
$$

AG

Ankit Gupta

Numerade Educator

Problem 59

Simplify. Do not use negative exponents in the answer.
$$
-10^{-2}
$$

AG

Ankit Gupta

Numerade Educator

Problem 60

Simplify. Do not use negative exponents in the answer.
$$
\left(5 a^{2} b^{6}\right)^{0}
$$

AG

Ankit Gupta

Numerade Educator

Problem 61

Simplify. Do not use negative exponents in the answer.
$$
(-3)^{2}-2-4 \cdot 6 \div 2 \cdot 3
$$

AG

Ankit Gupta

Numerade Educator

Problem 62

Simplify. Do not use negative exponents in the answer.
$$
\left(-3 x^{2} y^{-4}\right)\left(-2 x^{-7} y^{12}\right)
$$

AG

Ankit Gupta

Numerade Educator

Problem 63

Consider the following graph showing the amount of municipal solid waste recycled and the amount of municipal solid waste composted.
(IMAGE CANT COPY)
If the trends continue, will the amount of municipal solid waste composed ever equal the amount recycled? Why or why not?

AG

Ankit Gupta

Numerade Educator

Problem 64

Consider the following graph showing the amount of municipal solid waste recycled and the amount of municipal solid waste composted.
(IMAGE CANT COPY)
Suppose that the federal government wanted the amount composted to be the same as the amount recycled by $2030 .$ What would have to change? Do you think this can be achieved? Why or why not?

AG

Ankit Gupta

Numerade Educator

Problem 65

For each of the following conditions, write a system of equations. Answers may vary.
a) $(5,1)$ is a solution.
b) There is no solution.
c) There is an infinite number of solutions.

AG

Ankit Gupta

Numerade Educator

Problem 66

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

AG

Ankit Gupta

Numerade Educator

Problem 67

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

Cory Kuzinski

Numerade Educator

Problem 68

Translate to a system of equations. Do not solve.
Tyler is twice as old as his son. Ten years ago, Tyler was three times as old as his son. How old are they now?

AG

Ankit Gupta

Numerade Educator

Problem 69

Translate to a system of equations. Do not solve.
Dell and Juanita are mathematics professors at a state university. Together, they have 46 years of service. Two years ago, Dell had taught 2.5 times as many years as Juanita. How long has each taught at the university?

AG

Ankit Gupta

Numerade Educator

Problem 70

Translate to a system of equations. Do not solve.
A piece of posterboard has a perimeter of 156 in. If you cut 6 in. off the width, the length becomes four times the width. What are the dimensions of the original piece of posterboard?
(IMAGE CANT COPY)

AG

Ankit Gupta

Numerade Educator

Problem 71

Translate to a system of equations. Do not solve.
A nontoxic scouring powder is made up of 4 parts baking soda and 1 part vinegar. How much of each ingredient is needed for a 16 -oz mixture?

AG

Ankit Gupta

Numerade Educator

Problem 72

Solve Exercise 41 graphically.

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Ankit Gupta

Numerade Educator

Problem 73

Solve Exercise 44 graphically.

AG

Ankit Gupta

Numerade Educator

Problem 74

Solve graphically.
$$
\begin{aligned}
y &=|x| \\
3 y-x &=8
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 75

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

Cory Kuzinski

Numerade Educator

Problem 76

Use a graphing calculator to solve each system of linear equations for $x$ and $y .$ Round all coordinates to the nearest hundredth.
$$
\begin{aligned}
&y=8.23 x+2.11\\
&y=-9.11 x-4.66
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 77

Use a graphing calculator to solve each system of linear equations for $x$ and $y .$ Round all coordinates to the nearest hundredth.
$$
\begin{array}{l}
{y=-3.44 x-7.72} \\
{y=4.19 x-8.22}
\end{array}
$$

AG

Ankit Gupta

Numerade Educator

Problem 78

Use a graphing calculator to solve each system of linear equations for $x$ and $y .$ Round all coordinates to the nearest hundredth.
$$
\begin{aligned}
&14.12 x+7.32 y=2.98\\
&21.88 x-6.45 y=-7.22
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 79

Use a graphing calculator to solve each system of linear equations for $x$ and $y .$ Round all coordinates to the nearest hundredth.
$$
\begin{aligned}
5.22 x-8.21 y &=-10.21 \\
-12.67 x+10.34 y &=12.84
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 80

Solve graphically using the following grid:
$$
\begin{aligned}
2 x-3 y &=0 \\
-4 x+3 y &=-1
\end{aligned}
$$
(GRAPH CANT COPY)

AG

Ankit Gupta

Numerade Educator

Problem 81

Refer to the graph on the chapter opener on $\mathrm{p} .149 .$ Find data for the amount of municipal solid waste recovered for recycling and the amount recovered for composting for a recent year. Do the trends continue? If any new trends emerge, try to find reasons for the change.

AG

Ankit Gupta

Numerade Educator