College Algebra

Ron Larson, David C. Falvo

Chapter 6

Systems of Equations and Inequalities - all with Video Answers

Educators

Section 1

Linear and Nonlinear Systems of Equations

Problem 1

A set of two or more equations in two or more variables is called a _____ of _____ .

DD

Daniel Dore

Community College of the Air Force

Problem 2

A _____ of a system of equations is an ordered pair that satisfies each equation in the system.

Erika Bustos

Numerade Educator

Problem 3

Finding the set of all solutions to a system of equations is called _____ the system of equations.

Chris Wojturski

Chris Wojturski

Numerade Educator

Problem 4

The first step in solving a system of equations by the method of _____ is to solve one of the equations for one variable in terms of the other variable.

Ivan Kochetkov

Numerade Educator

Problem 5

Graphically, the solution of a system of two equations is the _____ of _____ of the graphs of the two equations.

Ivan Kochetkov

Numerade Educator

Problem 6

In business applications, the point at which the revenue equals costs is called the _____ point.

Ivan Kochetkov

Numerade Educator

Problem 7

Determine whether each ordered pair is a solution of the system of equations.
$\left\{\begin{array}{l}2 x-y=4 \\ 8 x+y=-9\end{array}\right.$
(a) $(0,-4)$ (b) $(-2,7)$ (c) $\left(\frac{3}{2},-1\right)$ (d) $\left(-\frac{1}{2},-5\right)$

Erika Bustos

Numerade Educator

Problem 8

Determine whether each ordered pair is a solution of the system of equations.
$\left\{\begin{array}{l}4 x^2+y=3 \\ -x-y=11\end{array}\right.$
(a) $(2,-13)$ (b) $(2,-9)$ (c) $\left(-\frac{3}{2},-\frac{31}{3}\right)$ (d) $\left(-\frac{2}{4},-\frac{37}{4}\right)$

Swati Agarwal

Numerade Educator

Problem 9

Determine whether each ordered pair is a solution of the system of equations.
$\left\{\begin{aligned} y & =-4 e^x \\ 7 x-y & =4\end{aligned}\right.$
(a) $(-4,0)$ (b) $(0,-4)$ (c) $(0,-2)$ (d) $(-1,-3)$

Charles Carter

Numerade Educator

Problem 10

Determine whether each ordered pair is a solution of the system of equations.
$\left\{\begin{aligned}-\log x+3 & =y \\ \frac{1}{9} x+y & =\frac{28}{9}\end{aligned}\right.$
(a) $\left(9, \frac{37}{9}\right)$ (b) $(10,2)$ (c) $(1,3)$ (d) $(2,4)$

AG

Ankit Gupta

Numerade Educator

Problem 11

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{r}2 x+y=6 \\ -x+y=0\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 12

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{l}x-4 y=-11 \\ x+3 y=3\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 13

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{r}x-y=-4 \\ x^2-y=-2\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 14

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{r}3 x+y=2 \\ x^3-2+y=0\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 15

Solve the system by the method of substitution. Check your solution(s) graphically.
. $\left\{\begin{aligned}-\frac{1}{2} x+y & =-\frac{5}{2} \\ x^2+y^2 & =25\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 16

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{r}x+y=0 \\ x^3-5 x-y=0\end{array}\right.$

Trinity Steen

Numerade Educator

Problem 17

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{aligned} x^2+y & =0 \\ x^2-4 x-y & =0\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 18

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{l}y=-2 x^2+2 \\ y=2\left(x^4-2 x^2+1\right)\end{array}\right.$

Alisa Lu

Numerade Educator

Problem 19

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{l}y=x^3-3 x^2+1 \\ y=x^2-3 x+1\end{array}\right.$

Trinity Steen

Numerade Educator

Problem 20

Solve the system by the method of substitution. Check your solution(s) graphically.
$\left\{\begin{array}{l}y=x^3-3 x^2+4 \\ y=-2 x+4\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 21

Solve the system by the method of substitution.
$\left\{\begin{aligned} x-y & =2 \\ 6 x-5 y & =16\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 22

Solve the system by the method of substitution.
$\left\{\begin{aligned} x+4 y= & 3 \\ 2 x-7 y= & -24\end{aligned}\right.$

Erika Bustos

Numerade Educator

Problem 23

Solve the system by the method of substitution.
$\left\{\begin{array}{l}2 x-y+2=0 \\ 4 x+y-5=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 24

Solve the system by the method of substitution.
$\left\{\begin{aligned} 6 x-3 y-4 & =0 \\ x+2 y-4 & =0\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 25

Solve the system by the method of substitution.
$\left\{\begin{array}{l}1.5 x+0.8 y=2.3 \\ 0.3 x-0.2 y=0.1\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 26

Solve the system by the method of substitution.
$\left\{\begin{array}{l}0.5 x+3.2 y=9.0 \\ 0.2 x-1.6 y=-3.6\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 27

Solve the system by the method of substitution.
$\left\{\begin{aligned} \frac{1}{5} x+\frac{1}{2} y & =8 \\ x+y & =20\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 28

Solve the system by the method of substitution.
$\left\{\begin{array}{l}\frac{1}{2} x+\frac{3}{4} y=10 \\ \frac{3}{4} x-y=4\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 29

Solve the system by the method of substitution.
$\left\{\begin{array}{r}6 x+5 y=-3 \\ -x-\frac{5}{6} y=-7\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 30

Solve the system by the method of substitution.
$\left\{\begin{array}{r}-\frac{2}{3} x+y=2 \\ 2 x-3 y=6\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 31

Solve the system by the method of substitution.
$\left\{\begin{array}{l}x^2-y=0 \\ 2 x+y=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 32

Solve the system by the method of substitution.
$\left\{\begin{array}{r}x-2 y=0 \\ 3 x-y^2=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 33

Solve the system by the method of substitution.
$\left\{\begin{array}{r}x-y=-1 \\ x^2-y=-4\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 34

Solve the system by the method of substitution.
$\left\{\begin{array}{l}y=-x \\ y=x^3+3 x^2+2 x\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 35

Solve the system graphically.
$\left\{\begin{aligned}-x+2 y & =-2 \\ 3 x+y & =20\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 36

Solve the system graphically.
$\left\{\begin{array}{r}x+y=0 \\ 3 x-2 y=5\end{array}\right.$

Erika Bustos

Numerade Educator

Problem 37

Solve the system graphically.
$\left\{\begin{aligned} x-3 y & =-3 \\ 5 x+3 y & =-6\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 38

Solve the system graphically.
$\left\{\begin{aligned}-x+2 y & =-7 \\ x-y & =2\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 39

Solve the system graphically.
$\left\{\begin{array}{r}x+y=4 \\ x^2+y^2-4 x=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 40

Solve the system graphically.
$\left\{\begin{array}{r}-x+y=3 \\ x^2-6 x-27+y^2=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 41

Solve the system graphically.
$\left\{\begin{array}{r}x-y+3=0 \\ x^2-4 x+7=y\end{array}\right.$

Trinity Steen

Numerade Educator

Problem 42

Solve the system graphically.
$\left\{\begin{aligned} y^2-4 x+11 & =0 \\ -\frac{1}{2} x+y & =-\frac{1}{2}\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 43

Solve the system graphically.
$\left\{\begin{aligned} 7 x+8 y & =24 \\ x-8 y & =8\end{aligned}\right.$

Charles Carter

Numerade Educator

Problem 44

Solve the system graphically.
$\left\{\begin{array}{r}x-y=0 \\ 5 x-2 y=6\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 45

Solve the system graphically.
$\left\{\begin{array}{l}3 x-2 y=0 \\ x^2-y^2=4\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 46

Solve the system graphically.
$\left\{\begin{array}{r}2 x-y+3=0 \\ x^2+y^2-4 x=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 47

Solve the system graphically.
$\left\{\begin{aligned} x^2+y^2 & =25 \\ 3 x^2-16 y & =0\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 48

Solve the system graphically.
$\left\{\begin{aligned} x^2+y^2 & =25 \\ (x-8)^2+y^2 & =41\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 49

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$\left\{\begin{aligned} y & =e^x \\ x-y+1 & =0\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 50

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$\left\{\begin{aligned} y & =-4 e^{-x} \\ y+3 x+8 & =0\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 51

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$\left\{\begin{aligned} x+2 y & =8 \\ y & =\log _2 x\end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 52

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$\left\{\begin{aligned} y+2 & =\ln (x-1) \\ 3 y+2 x & =9\end{aligned}\right.$

Trinity Steen

Numerade Educator

Problem 53

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$\left\{\begin{array}{l}x^2+y^2=169 \\ x^2-8 y=104\end{array}\right.$

Charles Carter

Numerade Educator

Problem 54

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$\left\{\begin{array}{l}x^2+y^2=4 \\ 2 x^2-y=2\end{array}\right.$

Trinity Steen

Numerade Educator

Problem 55

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}y=2 x \\ y=x^2+1\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 56

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{aligned} x^2+y^2 & =25 \\ 2 x+y & =10\end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 57

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}x-2 y=4 \\ x^2-y=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 58

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}y=(x+1)^3 \\ y=\sqrt{x-1}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 59

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}y-e^{-x}=1 \\ y-\ln x=3\end{array}\right.$

Charles Carter

Numerade Educator

Problem 60

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}x^2+y=4 \\ e^x-y=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 61

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}y=x^4-2 x^2+1 \\ y=1-x^2\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 62

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{l}y=x^3-2 x^2+x-1 \\ y=-x^2+3 x-1\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 63

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{array}{r}x y-1=0 \\ 2 x-4 y+7=0\end{array}\right.$

Swati Agarwal

Numerade Educator

Problem 64

Solve the system graphically or algebraically. Explain your choice of method.
$\left\{\begin{aligned} x-2 y & =1 \\ y & =\sqrt{x-1}\end{aligned}\right.$

Swati Agarwal

Numerade Educator

Problem 65

Find the sales necessary to break even ( $R=C$ ) for the cost $C$ of producing $x$ units and the revenue $R$ obtained by selling $x$ units.
$C=8650 x+250,000, \quad R=9950 x$

Trinity Steen

Numerade Educator

Problem 66

Find the sales necessary to break even ( $R=C$ ) for the cost $C$ of producing $x$ units and the revenue $R$ obtained by selling $x$ units.
$C=5.5 \sqrt{x}+10,000, \quad R=3.29 x$

Yujie Wang

College of San Mateo

Problem 67

A small software company invests $$\$ 25,000$$ to produce a software package that will sell for $$\$ 69.95$$. Each unit can be produced for $$\$ 45.25$$.
(a) How many units must be sold to break even?
(b) How many units must be sold to make a profit of $\$ 100,000$ ?

Noah Musser

Numerade Educator

Problem 68

A small fast-food restaurant invests $$\$ 10,000$$ to produce a new food item that will sell for $$\$ 3.99$$. Each item can be produced for $$\$ 1.90$$.
(a) How many items must be sold to break even?
(b) How many items must be sold to make a profit of $$\$ 12,000$$ ?

Alisa Lu

Numerade Educator

Problem 69

The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week. At the same time, the weekly rentals for a newly released DVD of a horror film increased each week. Models that approximate the weekly rentals $R$ for each DVD are $$\begin{cases}R=360-24 x & \text { Animated film } \\ R=24+18 x & \text { Horror film }\end{cases}$$ where $x$ represents the number of weeks each DVD was in the store, with $x=1$ corresponding to the first week.
(a) After how many weeks will the rentals for the two movies be equal?
(b) Use a table to solve the system of equations numerically. Compare your result with that of part (a).

Charles Carter

Numerade Educator

Problem 70

The total weekly sales for a newly released portable media player (PMP) increased each week. At the same time, the total weekly sales for another newly released PMP decreased each week. Models that approximate the total weekly sales $S$ (in thousands of units) for each PMP are $$\begin{cases}S=15 x+50 & \text { PMP } 1 \\ S=-20 x+190 & \text { PMP } 2\end{cases}$$ where $x$ represents the number of weeks each PMP was in stores, with $x=0$ corresponding to the PMP sales on the day each PMP was first released in stores.
(a) After how many weeks will the sales for the two PMPs be equal?
(b) Use a table to solve the system of equations numerically. Compare your result with that of part (a).

Alisa Lu

Numerade Educator

Problem 71

You are offered two jobs selling dental supplies. One company offers a straight commission of $6 \%$ of sales. The other company offers a salary of $$\$ 500$$ per week plus $3 \%$ of sales. How much would you have to sell in a week in order to make the straight commission offer better?

Charles Carter

Numerade Educator

Problem 72

The supply and demand curves for a business dealing with wheat are
Supply: $p=1.45+0.00014 x^2$
Demand: $p=(2.388-0.007 x)^2$
where $p$ is the price in dollars per bushel and $x$ is the quantity in bushels per day. Use a graphing utility to graph the supply and demand equations and find the market equilibrium. (The market equilibrium is the point of intersection of the graphs for $x>0$.)

Alisa Lu

Numerade Educator

Problem 73

A total of $$\$ 25,000$$ is invested in two funds paying $6 \%$ and $8.5 \%$ simple interest. (The $6 \%$ investment has a lower risk.) The investor wants a yearly interest income of $$\$2000$$ from the two investments.
(a) Write a system of equations in which one equation represents the total amount invested and the other equation represents the $$\$ 2000$$ required in interest. Let $x$ and $y$ represent the amounts invested at $6 \%$ and $8.5 \%$, respectively.
(b) Use a graphing utility to graph the two equations in the same viewing window. As the amount invested at $6 \%$ increases, how does the amount invested at $8.5 \%$ change? How does the amount of interest income change? Explain.
(c) What amount should be invested at $6 \%$ to meet the requirement of $$\$ 2000$$ per year in interest?

Charles Carter

Numerade Educator

Problem 74

You are offered two different rules for estimating the number of board feet in a 16 -foot log. (A board foot is a unit of measure for lumber equal to a board 1 foot square and 1 inch thick.) The first rule is the Doyle Log Rule and is modeled by $V_1=(D-4)^2$, $5 \leq D \leq 40$, and the other is the Scribner Log Rule and is modeled by $V_2=0.79 D^2-2 D-4,5 \leq D \leq 40$, where $D$ is the diameter (in inches) of the $\log$ and $V$ is its volume (in board feet).
(a) Use a graphing utility to graph the two $\log$ rules in the same viewing window.
(b) For what diameter do the two scales agree?
(c) You are selling large logs by the board foot. Which scale would you use? Explain your reasoning.

AG

Ankit Gupta

Numerade Educator

Problem 75

The table shows the consumption $C$ (in trillions of Btus) of solar energy and wind energy in the United States from 1998 through 2006.
(a) Use the regression feature of a graphing utility to find a cubic model for the solar energy consumption data and a quadratic model for the wind energy consumption data. Let $t$ represent the year, with $t=8$ corresponding to 1998.
(b) Use a graphing utility to graph the data and the two models in the same viewing window.
(c) Use the graph from part (b) to approximate the point of intersection of the graphs of the models. Interpret your answer in the context of the problem.
(d) Describe the behavior of each model. Do you think the models can be used to predict consumption of solar energy and wind energy in the United States for future years? Explain.
(e) Use your school's library, the Internet, or some other reference source to research the advantages and disadvantages of using renewable energy.

Noah Musser

Numerade Educator

Problem 76

The table shows the populations $P$ (in millions) of Georgia, New Jersey, and North Carolina from 2002 through 2007.
(a) Use the regression feature of a graphing utility to find linear models for each set of data. Let $t$ represent the year, with $t=2$ corresponding to 2002 .
(b) Use a graphing utility to graph the data and the models in the same viewing window.
(c) Use the graph from part (b) to approximate any points of intersection of the graphs of the models. Interpret the points of intersection in the context of the problem.
(d) Verify your answers from part (c) algebraically.

Noah Musser

Numerade Educator

Problem 77

The table shows the average costs (in dollars) of one year's tuition for public and private universities in the United States from 2000 through 2006.
(a) Use the regression feature of a graphing utility to find a quadratic model $T_1$ for tuition at public universities and a linear model $T_2$ for tuition at private universities. Let $t$ represent the year, with $t=0$ corresponding to 2000 .
(b) Use a graphing utility to graph the data and the two models in the same viewing window.
(c) Use the graph from part (b) to determine the year after 2006 in which tuition at public universities will exceed tuition at private universities.
(d) Verify your answer from part (c) algebraically.

Noah Musser

Numerade Educator

Problem 78

Find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 56 meters and the length is 4 meters greater than the width.

Erika Bustos

Numerade Educator

Problem 79

Find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 280 centimeters and the width is 20 centimeters less than the length.

AG

Ankit Gupta

Numerade Educator

Problem 80

Find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 42 inches and the width is threefourths the length.

Erika Bustos

Numerade Educator

Problem 81

Find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 484 feet and the length is $4 \frac{1}{2}$ times the width.

Charles Carter

Numerade Educator

Problem 82

Find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 30.6 millimeters and the length is 2.4 times the width.

Mukesh Devi

Numerade Educator

Problem 83

What are the dimensions of a rectangular tract of land if its perimeter is 44 kilometers and its area is 120 square kilometers?

Ivan Kochetkov

Numerade Educator

Problem 84

What are the dimensions of an isosceles right triangle with a two-inch hypotenuse and an area of 1 square inch?

Ivan Kochetkov

Numerade Educator

Problem 85

Determine whether the statement is true or false. Justify your answer.
. In order to solve a system of equations by substitution, you must always solve for $y$ in one of the two equations and then back-substitute.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 86

Determine whether the statement is true or false. Justify your answer.
If a system consists of a parabola and a circle, then the system can have at most two solutions.

AG

Ankit Gupta

Numerade Educator

Problem 87

Use a graphing utility to graph $y_1=4-x$ and $y_2=x-2$ in the same viewing window. Use the zoom and trace features to find the coordinates of the point of intersection. What is the relationship between the point of intersection and the solution found in Example 1?

Charles Carter

Numerade Educator

Problem 88

Use a graphing utility to graph the two equations in Example 3, $y_1=3 x^2+4 x-7$ and $y_2=2 x+1$, in the same viewing window. How many solutions do you think this system has? Repeat this experiment for the equations in Example 4. How many solutions does this system have? Explain your reasoning.

Alisa Lu

Numerade Educator

Problem 89

When solving a system of equations by substitution, how do you recognize that the system has no solution?

Charles Carter

Numerade Educator

Problem 90

Consider the system of equations $$\left\{\begin{array}{l}a x+b y=c \\d x+e y=f\end{array}\right.$$
(a) Find values for $a, b, c, d, e$, and $f$ so that the system has one distinct solution.
(b) Explain how to solve the system in part (a) by the method of substitution and graphically.
(c) Write a brief paragraph describing any advantages of the method of substitution over the graphical method of solving a system of equations.

Katie Jenkins

Numerade Educator

Problem 91

Find equations of lines whose graphs intersect the graph of the parabola $y=x^2$ at (a) two points, (b) one point, and (c) no points. (There is more than one correct answer.) Use graphs to support your answers.

Charles Carter

Numerade Educator