Precalculus

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Dave Sobecki

Chapter 10

Solving Systems of Linear Equations Using Gauss–Jordan Elimination - all with Video Answers

Educators

Section 1

Systems of Linear Equations

Problem 1

Explain in your own words how to solve a system of two linear equations by graphing.

James Kiss

James Kiss

Numerade Educator

Problem 2

Explain in your own words how to solve a system of two linear equations by substitution.

James Kiss

Numerade Educator

Problem 3

Explain in your own words how to solve a system of two linear equations using elimination by addition.

James Kiss

Numerade Educator

Problem 4

Which of the three solving techniques is the best choice for a system of three equations? Why?

James Kiss

Numerade Educator

Problem 5

Can a system of two linear equations have exactly two solutions? Explain.

Joseph Lentino

Numerade Educator

Problem 6

Describe how the solution sets differ for systems of linear equations that are consistent, inconsistent, and dependent.

James Kiss

Numerade Educator

Problem 7

Match each system in Problems $7-10$ with one of the following graphs, and use the graph to solve the system.
$$
\begin{array}{r}
2 x-4 y=8 \\
x-2 y=0
\end{array}
$$

James Kiss

Numerade Educator

Problem 8

Match each system with one of the following graphs, and use the graph to solve the system.
$$
\begin{array}{l}
x+y=3 \\
x-2 y=0
\end{array}
$$

Landon Basham

Numerade Educator

Problem 9

Match each system with one of the following graphs, and use the graph to solve the system.
$$
\begin{array}{l}
2 x-y=5 \\
3 x+2 y=-3
\end{array}
$$

Landon Basham

Numerade Educator

Problem 10

Match each system with one of the following graphs, and use the graph to solve the system.
$$
\begin{array}{l}
4 x-2 y=10 \\
2 x-y=5
\end{array}
$$

Landon Basham

Numerade Educator

Problem 11

Solve the system of equations in Problems $11-46 .$
$$
\begin{array}{l}
x+y=7 \\
x-y=3
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 12

Solve the system of equations.
$$
\begin{array}{l}
x-y=2 \\
x+y=4
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 13

Solve the system of equations.
$$
\begin{array}{l}
3 x-2 y=12 \\
7 x+2 y=8
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 14

Solve the system of equations.
$$
\begin{array}{r}
3 x-y=2 \\
x+2 y=10
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 15

Solve the system of equations.
$$
\begin{array}{lr}
3 u+5 v= & 15 \\
6 u+10 v= & -30
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 16

Solve the system of equations.
$$
\begin{aligned}
m+2 n &=4 \\
2 m+4 n &=-8
\end{aligned}
$$

Erika Bustos

Numerade Educator

Problem 17

Solve the system of equations.
$$
\begin{aligned}
3 x-y &=-2 \\
-9 x+3 y &=6
\end{aligned}
$$

Erika Bustos

Numerade Educator

Problem 18

Solve the system of equations.
$$
\begin{array}{l}
2 x-8 y=10 \\
8 x-32 y=40
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 19

Solve the system of equations.
$$
\begin{array}{l}
x-y=4 \\
x+3 y=12
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 20

Solve the system of equations.
$$
\begin{array}{l}
3 x-y=7 \\
2 x+3 y=1
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 21

Solve the system of equations.
$$
\begin{array}{l}
4 x+3 y=26 \\
3 x-11 y=-7
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 22

Solve the system of equations.
$$
\begin{array}{rr}
9 x-3 y= & 24 \\
11 x+2 y= & 1
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 23

Solve the system of equations.
$$
\begin{array}{l}
7 m+12 n=-1 \\
5 m-3 n=7
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 24

Solve the system of equations.
$$
\begin{array}{rr}
3 p+8 q= & 4 \\
15 p+10 q= & -10
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 25

Solve the system of equations.
$$
\begin{array}{l}
y=0.08 x \\
y=100+0.04 x
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 26

Solve the system of equations.
$$
\begin{array}{l}
0.2 u-0.5 v=0.07 \\
0.8 u-0.3 v=0.79
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 27

Solve the system of equations.
$$
\begin{array}{l}
\frac{2}{5} x+\frac{3}{2} y=2 \\
\frac{7}{3} x-\frac{5}{4} y=-5
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 28

Solve the system of equations.
$$
\begin{array}{r}
5 x-2 y=8 \\
2 x+3 y=-10
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 29

Solve the system of equations.
$$
\begin{array}{rr}
-2.3 y+4.1 z= & -14.21 \\
10.1 y-2.9 z= & 26.15
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 30

Solve the system of equations.
$$
\begin{array}{l}
5.4 x+4.2 y=-12.9 \\
3.7 x+6.4 y=-4.5
\end{array}
$$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 31

Solve the system of equations.
$$
\begin{aligned}
-2 x &=2 \\
x-3 y &=2 \\
-x+2 y+3 z &=-7
\end{aligned}
$$

Erika Bustos

Numerade Educator

Problem 32

Solve the system of equations.
$$
\begin{aligned}
2 y+z &=-4 \\
x-3 y+2 z &=9 \\
-y &=3
\end{aligned}
$$

Erika Bustos

Numerade Educator

Problem 33

Solve the system of equations.
$$
\begin{array}{r}
2 y-z=2 \\
-4 y+2 z=1 \\
x-2 y+3 z=0
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 34

Solve the system of equations.
$$
\begin{array}{r}
x+y-z=3 \\
x-2 z=1 \\
y+z=2
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 35

Solve the system of equations.
$$
\begin{aligned}
x-3 y &=2 \\
2 y+z &=-1 \\
x-y+z &=1
\end{aligned}
$$

Erika Bustos

Numerade Educator

Problem 36

Solve the system of equations.
$$
\begin{aligned}
-4 x+3 y &=1 \\
8 x-6 y &=4 \\
2 x-4 y+3 z &=6
\end{aligned}
$$

Erika Bustos

Numerade Educator

Problem 37

Solve the system of equations.
$$
\begin{array}{rr}
2 x+\quad z=-5 \\
x-\quad 3 z=-6 \\
4 x+2 y-z=-9
\end{array}
$$

Erika Bustos

Numerade Educator

Problem 38

Solve the system of equations.
$$
\begin{array}{rr}
x-3 y+z= & 4 \\
-x+4 y-4 z= & 1 \\
2 x-y+5 z= & -3
\end{array}
$$

James Kiss

Numerade Educator

Problem 39

Solve the system of equations.
$$
\begin{array}{rr}
x-y+z= & 1 \\
2 x+y+z= & 6 \\
7 x-y+5 z= & 15
\end{array}
$$

James Kiss

Numerade Educator

Problem 40

Solve the system of equations.
$$
\begin{array}{rr}
2 x-y+3 z= & 7 \\
x+2 y-z= & -3 \\
3 x+y+2 z= & 2
\end{array}
$$

James Kiss

Numerade Educator

Problem 41

Solve the system of equations.
$$
\begin{array}{rr}
2 a+4 b+3 c= & -6 \\
a-3 b+2 c= & -15 \\
-a+2 b-c= & 9
\end{array}
$$

James Kiss

Numerade Educator

Problem 42

Solve the system of equations.
$$
\begin{array}{r}
3 u-2 v+3 w=11 \\
2 u+3 v-2 w=-5 \\
u+4 v-w=-5
\end{array}
$$

James Kiss

Numerade Educator

Problem 43

Solve the system of equations.
$$
\begin{array}{l}
2 x-3 y+3 z=-5 \\
3 x+2 y-5 z=34 \\
5 x-4 y-2 z=23
\end{array}
$$

James Kiss

Numerade Educator

Problem 44

Solve the system of equations.
$$
\begin{array}{rr}
x+2 y+z= & 2 \\
-2 x+3 y-2 z= & -3 \\
x-5 y+z= & 2
\end{array}
$$

James Kiss

Numerade Educator

Problem 45

Solve the system of equations.
$$
\begin{array}{rr}
-x+2 y-z= & -4 \\
2 x+5 y-4 z= & -16 \\
x+y-z= & -4
\end{array}
$$

James Kiss

Numerade Educator

Problem 46

Solve the system of equations.
$$
\begin{array}{rr}
x-8 y+2 z= & -1 \\
x-3 y+z= & 1 \\
2 x-11 y+3 z= & 2
\end{array}
$$

James Kiss

Numerade Educator

Problem 47

In Problems 47 and 48 , solve each system for $p$ and $q$ in terms of $x$ and $y .$ Explain how you could check your solution and then perform the check.
$$
\begin{array}{l}
x=2+p-2 q \\
y=3-p+3 q
\end{array}
$$

Lauren Shelton

Numerade Educator

Problem 48

Solve each system for $p$ and $q$ in terms of $x$ and $y .$ Explain how you could check your solution and then perform the check.
$$
\begin{array}{l}
x=-1+2 p-q \\
y=4-p+q
\end{array}
$$

Landon Basham

Numerade Educator

Problem 49

Problems 49 and 50 refer to the system
$$
\begin{array}{l}
a x+b y=h \\
c x+d y=k
\end{array}
$$
where $x$ and $y$ are variables and $a, b, c, d, h,$ and $k$ are real constants.
Solve the system for $x$ and $y$ in terms of the constants $a, b, c, d$, $h,$ and $k .$ Clearly state any assumptions you must make about the constants during the solution process.

Landon Basham

Numerade Educator

Problem 50

Refer to the system $$ \begin{array}{l} a x+b y=h \\ c x+d y=k \end{array} $$ where $x$ and $y$ are variables and $a, b, c, d, h,$ and $k$ are real constants.
Discuss the nature of solutions to systems that do not satisfy the assumptions you made in Problem $49 .$

Landon Basham

Numerade Educator

Problem 51

AIRSPEED It takes a private airplane 8.75 hours to make the 2,100 -mile flight from Atlanta to Los Angeles and 5 hours to make the return trip. Assuming that the wind blows at a constant rate from Los Angeles to Atlanta, find the airspeed of the plane and the wind rate.

Landon Basham

Numerade Educator

Problem 52

AIRSPEED A plane carries enough fuel for 20 hours of flight at an airspeed of 150 miles per hour. How far can it fly into a $30 \mathrm{mph}$ headwind and still have enough fuel to return to its starting point? (This distance is called the point of no return.)

Landon Basham

Numerade Educator

Problem 53

RATE-TIME A crew of eight can row 20 kilometers per hour in still water. The crew rows upstream and then returns to its starting point in 15 minutes. If the river is flowing at $2 \mathrm{~km} / \mathrm{h}$, how far upstream did the crew row?
RATE-TIME A crew of eight can row 20 kilometers per hour in still water. The crew rows upstream and then returns to its starting point in 15 minutes. If the river is flowing at $2 \mathrm{~km} / \mathrm{h}$, how far upstream did the crew row?

Landon Basham

Numerade Educator

Problem 54

RATE-TIME It takes a boat 2 hours to travel 20 miles down a river and 3 hours to return upstream to its starting point. What is the rate of the current in the river?

Erika Bustos

Numerade Educator

Problem 55

BUSINESS A company that supplies bulk candy to bakeries has one batch of chocolate chips that are $50 \%$ dark chocolate and $50 \%$ milk chocolate. They have another batch that is $80 \%$ dark chocolate and $20 \%$ milk chocolate. One of their customers sends in a rush order for $100 \mathrm{lb}$ of a mix that is $68 \%$ dark chocolate. How many pounds from each batch should be mixed to meet this order?

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 56

BUSINESS A jeweler has two bars of gold alloy in stock, one of 12 carats and the other of 18 carats ( 24 -carat gold is pure gold, 12 carat is $\frac{12}{24}$ pure, 18 -carat gold is $\frac{18}{24}$ pure, and so on). How many grams of each alloy must be mixed to obtain 10 grams of 14 -carat gold?

Landon Basham

Numerade Educator

Problem 57

BREAK-EVEN ANALYSIS It costs a small recording company $\$ 17,680$ to prepare a compact disc. This is a one-time fixed cost that covers recording, package design, and so on. Variable costs, including such things as manufacturing, marketing, and royalties, are $\$ 4,60$ per $\mathrm{CD}$. If the $\mathrm{CD}$ is sold to music shops for $\$ 8$ each, how many must be sold for the company to break even?

Landon Basham

Numerade Educator

Problem 58

FINANCE Suppose you have $\$ 12,000$ to invest. If part is invested at $10 \%$ and the rest at $15 \%$, how much should be invested at each rate to yield $12 \%$ on the total amount invested?

Landon Basham

Numerade Educator

Problem 59

PRODUCTION A supplier for the electronics industry manufactures keyboards and screens for graphing calculators at plants in Mexico and Taiwan. The hourly production rates at each plant are given in the table. How many hours should each plant be operated to fill an order for exactly 4,000 keyboards and exactly 4,000 screens?
\begin{tabular}{lcc}
Plant & Keyboards & Screens \\
\hline Mexico & 40 & 32 \\
Taiwan & 20 & 32 \\
\hline
\end{tabular}

Landon Basham

Numerade Educator

Problem 60

PRODUCTION A company produces Italian sausages and bratwursts at plants in Green Bay and Sheboygan. The hourly production rates at each plant are given in the table. How many hours should each plant be operated to exactly fill an order for 62,250 Italian sausages and 76,500 bratwursts?
\begin{tabular}{lcc}
Plant & Italian Sausage & Bratwurst \\
\hline Green Bay & 800 & 800 \\
Sheboygan & 500 & 1,000 \\
\hline
\end{tabular}

Landon Basham

Numerade Educator

Problem 61

SUPPLY AND DEMAND Suppose the supply and demand equations for printed T-shirts in a resort town for a particular week are
$$
\begin{array}{ll}
p=0.007 q+3 & \text { Supply equation } \\
p=-0.018 q+15 & \text { Demand equation }
\end{array}
$$
where $p$ is the price in dollars and $q$ is the quantity.
(A) Find the supply and the demand (to the nearest unit) if T-shirts are priced at $\$ 4$ each. Discuss the stability of the T-shirt market at this price level.
(B) Find the supply and the demand (to the nearest unit) if T-shirts are priced at $\$ 8$ each. Discuss the stability of the T-shirt market at this price level.
(C) Find the equilibrium price and quantity.
(D) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve.

Landon Basham

Numerade Educator

Problem 62

SUPPLY AND DEMAND Suppose the supply and demand equations for printed baseball caps in a resort town for a particular week are
$$
\begin{array}{ll}
p=0.006 q+2 & \text { Supply equation } \\
p=-0.014 q+13 & \text { Demand equation }
\end{array}
$$
where $p$ is the price in dollars and $q$ is the quantity in hundreds.
(A) Find the supply and the demand (to the nearest unit) if baseball caps are priced at $\$ 4$ each. Discuss the stability of the baseball cap market at this price level.
(B) Find the supply and the demand (to the nearest unit) if baseball caps are priced at $\$ 8$ each. Discuss the stability of the baseball cap market at this price level.
(C) Find the equilibrium price and quantity.
(D) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve.

Landon Basham

Numerade Educator

Problem 63

SUPPLY AND DEMAND At $\$ 0.60$ per bushel, the daily supply for wheat is 450 bushels and the daily demand is 645 bushels. When the price is raised to $\$ 0.90$ per bushel, the daily supply increases to 750 bushels and the daily demand decreases to 495 bushels. Assume that the supply and demand equations are linear.
(A) Find the supply equation.
(B) Find the demand equation.
(C) Find the equilibrium price and quantity.

Landon Basham

Numerade Educator

Problem 64

SUPPLY AND DEMAND At $\$ 1.40$ per bushel, the daily supply for soybeans is 1,075 bushels and the daily demand is 580 bushels. When the price falls to $\$ 1.20$ per bushel, the daily supply decreases to 575 bushels and the daily demand increases to 980 bushels. Assume that the supply and demand equations are linear.
(A) Find the supply equation.
(B) Find the demand equation.
(C) Find the equilibrium price and quantity.

Landon Basham

Numerade Educator

Problem 65

EARTH SCIENCE An earthquake emits a primary wave and a secondary wave. Near the surface of the Earth the primary wave travels at about 5 miles per second and the secondary wave at about 3 miles per second. From the time lag between the two waves arriving at a given receiving station, it is possible to estimate the distance to the quake. (The epicenter can be located by obtaining distance bearings at three or more stations.) Suppose a station measured a time difference of 16 seconds between the arrival of the two waves. How long did each wave travel, and how far was the earthquake from the station?

Landon Basham

Numerade Educator

Problem 66

EARTH SCIENCE A ship using sound-sensing devices above and below water recorded a surface explosion 6 seconds sooner by its underwater device than its above-water device. Sound travels in air at about 1,100 feet per second and in seawater at about 5,000 feet per second.
(A) How long did it take each sound wave to reach the ship?
(B) How far was the explosion from the ship?

Landon Basham

Numerade Educator

Problem 67

PRODUCTION SCHEDULING A company manufactures three products; lawn mowers, snowblowers, and chain saws. The labor, material, and shipping costs for manufacturing one unit of each product are given in the table. The weekly allocations for labor, materials, and shipping are $\$ 35,000, \$ 50,000,$ and $\$ 20,000,$ respectively. How many of each type of product should be manufactured each week in order to exactly use the weekly allocations? \begin{tabular}{lccc}
Product & Labor & Materials & Shipping \\
\hline Lawn mower & $\$ 20$ & $\$ 35$ & $\$ 15$ \\
Snowblower & $\$ 30$ & $\$ 50$ & $\$ 25$ \\
Chain saw & $\$ 45$ & $\$ 40$ & $\$ 10$ \\
\hline
\end{tabular}

Cinsy Krehbiel

Numerade Educator

Problem 68

PRODUCTION SCHEDULING A company manufactures three products; desk chairs, file cabinets, and printer stands. The labor, material, and shipping costs for manufacturing one unit of each product are given in the table. The weekly allocations for labor, materials, and shipping are $\$ 21,100, \$ 31,500,$ and $\$ 11,900,$ respectively. How many of each type of product should be manufactured each week in order to exactly use the weekly allocations? 1 \begin{tabular}{lccc}
Product & Desk Chair & File Cabinet & Printer Stand \\
\hline Labor & $\$ 30$ & $\$ 35$ & $\$ 40$ \\
Materials & $\$ 45$ & $\$ 60$ & $\$ 55$ \\
Shipping & $\$ 25$ & $\$ 20$ & $\$ 15$ \\
\hline
\end{tabular}

Cinsy Krehbiel

Numerade Educator

Problem 69

PRODUCTION SCHEDULING A company has plants located in Michigan, New York, and Ohio where it manufactures laptop computers, desktop computers, and servers. The number of units of each product that can be produced per day at each plant are given in the table below. The company has orders for 2,150 laptop computers, 2,300 desktop computers, and 2,500 servers. How many days should the company operate each plant in order to exactly fill these orders? \begin{tabular}{lccc}
Plant & Michigan & New York & Ohio \\
\hline Laptop & 10 & 70 & 60 \\
Desktop & 20 & 50 & 80 \\
Server & 40 & 30 & 90 \\
\hline
\end{tabular}

Cinsy Krehbiel

Numerade Educator

Problem 70

PRODUCTION SCHEDULING A company has plants located in Maine, Utah, and Oregon where it manufactures stoves, refrigerators, and dishwashers. The number of units of each product that can be produced per day at each plant are given in the table. The company has orders for 1,500 stoves, 2,350 refrigerators, and 2,400 dishwashers.
How many days should the company operate each plant in order to exactly fill these orders? Set up a system of equations whose solution would answer this question and solve the system. \begin{tabular}{lccc}
Plant & Stoves & Refrigerators & Dishwashers \\
\hline Maine & 30 & 70 & 60 \\
Utah & 20 & 50 & 50 \\
Oregon & 40 & 30 & 40 \\
\hline
\end{tabular}

Cinsy Krehbiel

Numerade Educator

Problem 71

INVESTMENT Due to recent volatility in the stock market, Catalina's financial advisor suggests that she reallocate $\$ 70,000$ of her retirement fund to bonds. He recommends a mix of treasury bonds earning $4 \%$ annually, municipal bonds earning $3.5 \%$ annually, and corporate bonds earning $4.5 \%$ annually. For tax reasons, he also recommends that the amount invested in treasury bonds should be equal to the sum of the amount invested in the other categories. If Catalina follows these recommendations, and the goal is to produce $\$ 2,900$ in annual interest income, how much will she invest in each of the three types of bonds?

Cinsy Krehbiel

Numerade Educator

Problem 72

INVESTMENT When the real estate market begins to rebound, Catalina (see Problem 71 ) decides to reallocate her investment mix. At this point, her investment has grown to $\$ 76,000$. She'll leave some money in treasury and corporate bonds, but will replace municipal bonds with a real estate investment trust that guarantees a $6.5 \%$ annual return. If she plans to leave as much in treasury bonds as the sum of the other two investments, how much should she invest in each to reach her new goal of earning an annual interest income of $\$ 3,600 ?$

Heather Zimmers

Heather Zimmers

Numerade Educator