Precalculus: Graphs and Models

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

Chapter 8

Modeling with Systems of Equations and Inequalities - all with Video Answers

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Section 1

Systems of Linear Equations in Two Variables

Problem 1

Describe how the solution sets for a consistent system, for an inconsistent system, and for a dependent system differ.

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Problem 2

Can a system have exactly two solutions? Explain.

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Problem 3

What is a parameter?

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Problem 4

Explain in your own words how to solve a system using a graphing calculator.

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Problem 5

Explain in your own words how to solve a system using substitution.

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Problem 6

Explain in your own words how to solve a system using elimination.

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Problem 7

Match each system with one of the following graphs, and use the graph to solve the system.
(Check your book to see graph)
$$\begin{array}{r}2 x-4 y=8 \\x-2 y=0\end{array}$$

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Problem 8

Match each system with one of the following graphs, and use the graph to solve the system.
(Check your book to see graph)
$$\begin{aligned}&x+y=3\\&x-2 y=0\end{aligned}$$

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Problem 9

Match each system with one of the following graphs, and use the graph to solve the system.
(Check your book to see graph)
$$\begin{aligned}&2 x-y=5\\&3 x+2 y=-3\end{aligned}$$

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Problem 10

Match each system with one of the following graphs, and use the graph to solve the system.
(Check your book to see graph)
$$\begin{aligned}&4 x-2 y=10\\&2 x-y=5\end{aligned}$$

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Problem 11

Solve Problems.
$x+y=7$
$x-y=3$

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Problem 12

Solve Problems.
$x-y=2$
$x+y=4$

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Problem 13

Solve Problems.
$3 x-2 y=12$
$7 x+2 y=8$

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Problem 14

Solve Problems.
$3 x-y=2$
$x+2 y=10$

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Problem 15

Solve Problems.
$3 u+5 v=15$
$6 u+10 v=-30$

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Problem 16

Solve Problems.
$\begin{array}{rr}m+2 n= & 4 \\ 2 m+4 n= & -8\end{array}$

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Problem 17

Solve Problems.
$y=2 x+3$
$y=3 x-5$

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Problem 18

Solve Problems.
$y=x+4$
$y=5 x-8$

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Problem 19

Solve Problems.
$x-y=4$
$x+3 y=12$

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Problem 20

Solve Problems.
$2 x-y=3$
$x+2 y=14$

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Problem 21

Solve Problems.
$3 x-y=7$
$2 x+3 y=1$

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Problem 22

Solve Problems.
$\begin{array}{rr}2 x+y= & 6 \\ x-y= & -3\end{array}$

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Problem 23

Solve Problems.
$4 x+3 y=26$
$3 x-11 y=-7$

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Problem 24

Solve Problems.
$9 x-3 y=24$
$11 x+2 y=1$

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Problem 25

Solve Problems.
$7 m+12 n=-1$
$5 m-3 n=7$

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Problem 26

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

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Problem 27

Solve Problems.
$y=0.08 x$
$y=100+0.04 x$

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Problem 28

Solve Problems.
$y=0.07 x$
$y=80+0.05 x$

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Problem 29

Solve Problems.
$0.2 u-0.5 v=0.07$
$0.8 u-0.3 v=0.79$

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Problem 30

Solve Problems.
$0.3 s-0.6 t=0.18$
$0.5 s-0.2 t=0.54$

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Problem 31

Solve Problems.
$\frac{2}{5} x+\frac{3}{2} y=2$
$\frac{7}{3} x-\frac{5}{4} y=-5$

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Problem 32

Solve Problems.
$\frac{7}{2} x-\frac{5}{6} y=10$
$\frac{2}{5} x+\frac{4}{3} y=6$

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Problem 33

Solve Problems.
$2 x-3 y=-5$
$3 x+4 y=13$

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Problem 34

Solve Problems.
$7 x-3 y=20$
$5 x+2 y=8$

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Problem 35

Solve Problems.
$3.5 x-2.4 y=0.1$
$2.6 x-1.7 y=-0.2$

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Problem 36

Solve Problems.
$5.4 x+4.2 y=-12.9$
$3.7 x+6.4 y=-4.5$

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Problem 37

Solve system for p and q in terms of $x$ and $y .$ Explain how you could check your solution and then perform the check.
$$\begin{aligned}&x=2+p-2 q\\&y=3-p+3 q\end{aligned}$$

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Problem 38

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

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Problem 39

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.

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Problem 40

Discuss the nature of solutions to systems that do not satisfy the assumptions you made in Problem 39

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Problem 41

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.

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Problem 42

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 mph headwind and still have enough fuel to return to its starting point? (This distance is called the point of no return.)

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Problem 43

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?

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Problem 44

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?

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Problem 45

A chemist has two solutions of hydrochloric acid in stock: a $50 \%$ solution and an $80 \%$ solution. How much of each should be used to obtain 100 milliliters of a $68 \%$ solution?

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Problem 46

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?

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Problem 47

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 CD. If the CD is sold to music shops for $\$ 8$ each, how many must be sold for the company to break even?

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Problem 48

A videocassette manufacturer has determined that its weekly cost equation is $C=3,000+10 x$ where $x$ is the number of cassettes produced and sold each week. If cassettes are sold for $\$ 15$ each to distributors, how many must be sold each week for the manufacturer to break even?

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Problem 49

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?

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Problem 50

An investor has $\$ 20,000$ to invest. If part is invested at $8 \%$ and the rest at $12 \%,$ how much should be invested at each rate to yield $11 \%$ on the total amount invested?

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Problem 51

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?
$$\begin{array}{lcc}\text { Plant } & \text { Keyboards } & \text { Screens } \\\hline \text { Mexico } & 40 & 32 \\\text { Taiwan } & 20 & 32 \\\hline\end{array}$$

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Problem 52

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?
$$\begin{array}{lcc}\text { Plant } & \text { Italian sausage } & \text { Bratwurst } \\\hline \text { Green Bay } & 800 & 800 \\\text { Sheboygan } & 500 & 1,000 \\\hline\end{array}$$

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Problem 53

Animals in an experiment are to be kept on a strict diet. Each animal is to receive, among other things, 20 grams of protein and 6 grams of fat. The laboratory technician is able to purchase two food mixes of the following compositions: Mix A has $10 \%$ protein and $6 \%$ fat, $\operatorname{mix} \mathrm{B}$ has $20 \%$ protein and $2 \%$ fat. How many grams of each mix should be used to obtain the right diet for a single animal?

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Problem 54

A fruit grower can use two types of fertilizer in an orange grove, brand $\mathrm{A}$ and brand $\mathrm{B}$. Each bag of brand $\mathrm{A}$ contains 8 pounds of nitrogen and 4 pounds of phosphoric acid. Each bag of brand $\mathrm{B}$ contains 7 pounds of nitrogen and 7 pounds of phosphoric acid. Tests indicate that the grove needs 720 pounds of nitrogen and 500 pounds of phosphoric acid. How many bags of each brand should be used to provide the required amounts of nitrogen and phosphoric acid?

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Problem 55

Suppose the supply and demand equations for printed T-shirts in a resort town for a particular week are
$$p=0.007 q+3 \quad \text { Supply equation }$$
$$p=-0.018 q+15 \quad \text { Demand equation }$$
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.

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Problem 56

Suppose the supply and demand equations for printed baseball caps in a resort town for a particular week are
$$p=0.006 q+2 \quad \text { Supply equation }$$
$$p=-0.014 q+13 \quad \text { Demand equation }$$
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.

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Problem 57

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.

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Problem 58

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.

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Problem 59

An object dropped off the top of a tall building falls vertically with constant acceleration. If $s$ is the distance of the object above the ground (in feet) $t$ seconds after its release, then $s$ and $t$ are related by an equation of the form
$$s=a+b t^{2}$$
where $a$ and $b$ are constants. Suppose the object is 180 feet above the ground 1 second after its release and 132 feet above the ground 2 seconds after its release.
(A) Find the constants $a$ and $b$
(B) How high is the building?
(C) How long does the object fall?

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Problem 60

Repeat Problem 59 if the object is 240 feet above the ground after 1 second and 192 feet above the ground after 2 seconds.

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Problem 61

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?

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Problem 62

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?

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