Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 10

Systems of Equations and Inequalities - all with Video Answers

Educators

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Section 1

Systems of Equations

Problem 1

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{aligned} x-y &=2 \\ 2 x+3 y &=9 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 2

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{array}{c}{2 x+y=7} \\ {x+2 y=2}\end{array}\right.$

Marcial Vargas

Numerade Educator

Problem 3

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{array}{l}{y=x^{2}} \\ {y=x+12}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 4

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{aligned} x^{2}+y^{2} &=25 \\ y &=2 x \end{aligned}\right.$

Sarah Lewites

Numerade Educator

Problem 5

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {x+y=0}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 6

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{aligned} x^{2}+y &=9 \\ x-y+3 &=0 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 7

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{aligned} x+y^{2} &=0 \\ 2 x+5 y^{2} &=75 \end{aligned}\right.$

Sarah Lewites

Numerade Educator

Problem 8

Use the substitution method to find all solutions of the system of equations.
$\left\{\begin{aligned} x^{2}-y &=1 \\ 2 x^{2}+3 y &=17 \end{aligned}\right.$

Sarah Lewites

Numerade Educator

Problem 9

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{array}{r}{x+2 y=5} \\ {2 x+3 y=8}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 10

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{array}{l}{4 x-3 y=11} \\ {8 x+4 y=12}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 11

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{array}{l}{x^{2}-2 y=1} \\ {x^{2}+5 y=29}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 12

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{array}{l}{3 x^{2}+4 y=17} \\ {2 x^{2}+5 y=2}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 13

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{array}{c}{3 x^{2}-y^{2}=11} \\ {x^{2}+4 y^{2}=8}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 14

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{aligned} 2 x^{2}+4 y &=13 \\ x^{2}-y^{2} &=\frac{7}{2} \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 15

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{aligned} x-y^{2}+3 &=0 \\ 2 x^{2}+y^{2}-4 &=0 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 16

Use the elimination method to find all solutions of the system of equations.
$\left\{\begin{array}{c}{x^{2}-y^{2}=1} \\ {2 x^{2}-y^{2}=x+3}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 17

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$\left\{\begin{array}{c}{2 x+y=-1} \\ {x-2 y=-8}\end{array}\right.$

Jake Zanazzi

Numerade Educator

Problem 18

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$\left\{\begin{array}{c}{x+y=2} \\ {2 x+y=5}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 19

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$\left\{\begin{array}{rr}{x^{2}+y=} & {8} \\ {x-2 y=} & {-6}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 20

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$\left\{\begin{array}{l}{x-y^{2}=-4} \\ {x-y=2}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 21

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$\left\{\begin{aligned} x^{2}+y &=0 \\ x^{3}-2 x-y &=0 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 22

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$\left\{\begin{aligned} x^{2}+y^{2} &=4 x \\ x &=y^{2} \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 23

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{y+x^{2}=4 x} \\ {y+4 x=16}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 24

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{x-y^{2}=0} \\ {y-x^{2}=0}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 25

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{x-2 y=2} \\ {y^{2}-x^{2}=2 x+4}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 26

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{y=4-x^{2}} \\ {y=x^{2}-4}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 27

Find all solutions of the system of equations.
$\left\{\begin{aligned} x-y &=4 \\ x y &=12 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 28

Find all solutions of the system of equations.
$\left\{\begin{aligned} x y &=24 \\ 2 x^{2}-y^{2}+4 &=0 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 29

Find all solutions of the system of equations.
$\left\{\begin{aligned} x^{2} y &=16 \\ x^{2}+4 y+16 &=0 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 30

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{x+\sqrt{y}=0} \\ {y^{2}-4 x^{2}=12}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 31

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{x^{2}+y^{2}=9} \\ {x^{2}-y^{2}=1}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 32

Find all solutions of the system of equations.
$\left\{\begin{aligned} x^{2}+2 y^{2} &=2 \\ 2 x^{2}-3 y &=15 \end{aligned}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 33

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{2 x^{2}-8 y^{3}=19} \\ {4 x^{2}+16 y^{3}=34}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 34

Find all solutions of the system of equations.
$\left\{\begin{aligned} x^{4}-y^{3} &=17 \\ 3 x^{4}+5 y^{3} &=53 \end{aligned}\right.$

Erin Kearney

Numerade Educator

Problem 35

Find all solutions of the system of equations.
$\left\{\begin{array}{c}{\frac{2}{x}-\frac{3}{y}=1} \\ {-\frac{4}{x}+\frac{7}{y}=1}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 36

Find all solutions of the system of equations.
$\left\{\begin{array}{l}{\frac{4}{x^{2}}+\frac{6}{y^{4}}=\frac{7}{2}} \\ {\frac{1}{x^{2}}-\frac{2}{y^{4}}=0}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 37

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{y=2 x+6} \\ {y=-x+5}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 38

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{y=-2 x+12} \\ {y=x+3}\end{array}\right.$

Erin Kearney

Numerade Educator

Problem 39

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{y=x^{2}+8 x} \\ {y=2 x+16}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 40

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{y=x^{2}-4 x} \\ {2 x-y=2}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 41

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {x+3 y=2}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 42

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{aligned} x^{2}+y^{2} &=17 \\ x^{2}-2 x+y^{2} &=13 \end{aligned}\right.$

Sarah Lewites

Numerade Educator

Problem 43

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{\frac{x^{2}}{9}+\frac{y^{2}}{18}=1} \\ {y=-x^{2}+6 x-2}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 44

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{x^{2}-y^{2}=3} \\ {y=x^{2}-2 x-8}\end{array}\right.$

AG

Ankit Gupta

Numerade Educator

Problem 45

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{x^{4}+16 y^{4}=32} \\ {x^{2}+2 x+y=0}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 46

Use the graphical method to find all solutions of the system of equations, correct to two decimal places.
$\left\{\begin{array}{l}{y=e^{x}+e^{-x}} \\ {y=5-x^{2}}\end{array}\right.$

Sarah Lewites

Numerade Educator

Problem 47

A rectangle has an area of 180 $\mathrm{cm}^{2}$ and a perimeter of 54 $\mathrm{cm} .$ What are its dimensions?

AS

Alexander Shashkov

Numerade Educator

Problem 48

A right triangle has an area of 84 $\mathrm{ft}^{2}$ and a hypotenuse 25 $\mathrm{ft}$ long. What are the lengths of its other two sides?

AG

Ankit Gupta

Numerade Educator

Problem 49

The perimeter of a rectangle is 70 and its diagonal is $25 .$ Find its length and width.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 50

A circular piece of sheet metal has a diameter of 20 in. The edges are to be cut off to form a rectangle of area 160 in $^{2}$ (see the figure). What are the dimensions of the rectangle?

James Kiss

Numerade Educator

Problem 51

A hill is inclined so that its “slope” is $\frac{1}{2}$ , as shown in the figure. We introduce a coordinate system with the origin at the base of the hill and with the scales on the axes measured in meters. A rocket is fired from the base of the hill in such a way that its trajectory is the parabola $y=-x^{2}+401 x$ . At what point does the rocket strike the hillside? How far is this point from the base of the hill (to the nearest $\mathrm{cm} )$ ?

Pawan Yadav

Numerade Educator

Problem 52

A rectangular piece of sheet metal with an area of 1200 $\mathrm{in}^{2}$ is to be bent into a cylindrical length of stovepipe having a volume of 600 $\mathrm{in}^{3} .$ What are the dimensions of the sheet metal?

Stephen Hobbs

Numerade Educator

Problem 53

The Global Positioning System determines the location of an object from its distances to satellites in orbit around the earth. In the simplified, two-dimensional situation shown in the figure, determine the coordinates of P from the fact that P is 26 units from satellite A and 20 units from satellite B.

AG

Ankit Gupta

Numerade Educator

Problem 54

On a sheet of graph paper, or using a graphing calculator, draw the parabola $y=x^{2} .$ Then draw the graphs of the linear equation $y=x+k$ on the same coordinate plane for various values of $k .$ Try to choose values of $k$ so that the line and the parabola intersect at two points for some of your $k^{\prime} \mathrm{s}$ , and not for others. For what value of $k$ is there exactly one intersection point? Use the results of $k$ is there exactly one make a conjecture about the values of $k$ for which the following system has two solutions, one solution, and no solution. Prove your conjecture.
$$\left\{\begin{array}{l}{y=x^{2}} \\ {y=x+k}\end{array}\right.$$

Erin Kearney

Numerade Educator

Problem 55

Follow the hints and solve the systems.
(a) $\left\{\begin{array}{cc}{\log x+\log y} & {=\frac{3}{2}} \\ {2 \log x-\log y} & {=0}\end{array} \quad[\text { Hint: Add the equations. }]\right.$
(b) $\left\{\begin{array}{ll}{2^{x}+2^{y}=10} & {\text { [Hint: Note that }} \\ {4^{x}+4^{y}=68} & {4^{x}=2^{2 x}=\left(2^{x}\right)^{2} . ]}\end{array}\right.$
(c) $\left\{\begin{array}{rlrl}{x-y} & {=3} & {} & {\text { [Hint: Factor the left side }} \\ {x^{3}-y^{3}} & {=387} & {} & {\text { of the second equation. } ]}\end{array}\right.$
(d) $\left\{\begin{array}{ll}{x^{2}+x y=1} & {\text { [Hint: Add the equations }} \\ {x y+y^{2}=3} & {\text { and factor the result. } ]}\end{array}\right.$

Dale Sanford

Numerade Educator