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6 Linear Equations: Getting Information from a Model

# 6 Linear Equations: Getting Information from a Model

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202

CHAPTER 2

Linear Functions and Models

the time x where these two graphs intersect. From the figure we see the height of the

graph of y = 200 + 5x reaches 10,000 when x is about 1950. So the pool is filled in

y

12,000

10,000

Pool is filled when

y=10,000 gallons

y=10,000

8000

We want to find the time x when the

pool is filled.

6000

y=200+5x

4000

Pool is filled in

2000

f i g u r e 1 Graphs of

y = 200 + 5x and y = 10,000

0

400

800

1200

1600

2000

2400 x

Solution 2 Algebraic

The pool is filled when the volume y is 10,000 gallons. Replacing y by 10,000 in the

equation y = 200 + 5x gives us a one-variable equation in the variable x. We solve

this equation for x, the time when the pool is filled.

y = 200 + 5x

10,000 = 200 + 5x

9800 = 5x

x =

Equation

Replace y by 10,000

Subtract 200 from each side

9800

5

x = 1960

Divide by 5, and switch sides

Calculator

So 1960 is a solution to the equation. This means that it takes 1960 minutes to fill the

pool. Notice that the algebraic solution gives us an exact answer, whereas the graphical solution is approximate.

NOW TRY EXERCISE 21

In Example 1 the solution is 1960 minutes. Such a large number of minutes is

better understood if it is stated in hours:

1960 minutes =

1960 minutes 1960 minutes hour

2

=

= 32 hours

minutes

60

minutes

3

60

hour

So it takes 32 hours and 40 minutes to fill the pool.

2

■ Models That Lead to Linear Equations

Recall that a model is a function that represents a real-world situation. Here we construct

models that lead to linear equations. (Getting information from these models requires us

to solve linear equations.) In constructing models, we use the following guidelines.

SECTION 2.6

Linear Equations: Getting Information from a Model

203

How to Construct a Model

Identify the varying quantity in the problem (the

independent variable) and give it a name, such as x.

2. Translate words to algebra. Express all the quantities given in the

problem in terms of the variable x.

3. Set up the model. Express the model algebraically as a function of the

variable x.

1. Choose the variable.

In the next example we construct a model involving simple interest. We use the

following simple interest formula, which gives the amount of interest I earned

when a principal P is deposited for t years at an interest rate r:

I = Prt

When using this formula, remember to convert r from a percentage to a decimal. For

example, in decimal form, 5% is 0.05. So at an interest rate of 5% the interest paid

on a \$1000 deposit over a 3-year period is I = Prt = 100010.05 2 132 = \$150.

e x a m p l e 2 Constructing and Using a Model (Interest)

Mary inherits \$100,000 and invests it in two one-year certificates of deposit. One certificate pays 6%, and the other pays 4 12 % simple interest annually.

(a) Construct a model for the total interest Mary earns in one year on her

investments.

(b) If Mary’s total interest is \$5025, how much money did she invest in each

certificate?

Solution

(a) Choose the variable. The variable in this problem is the amount that Mary

invests in each certificate. So let

x = amount invested at 6%

Translate words to algebra. Since Mary’s total inheritance is \$100,000

and she invests x dollars at 6%, it follows that she invested 100,000 - x at

4 12 % in the second certificate. Let’s translate all the information given in the

problem into the language of algebra.

In Words

In Algebra

Amount invested at 6%

Amount invested at 4 12%

Interest earned at 6%

Interest earned at 4 12%

x

100,000 - x

0.06x

0.0451100,000 - x2

Set up the model. We are now ready to set up the model. The function we

want gives the total interest Mary earns (the interest she earns at 6% plus the

interest she earns at 4 12 % ).

204

CHAPTER 2

Linear Functions and Models

Interest earned

at 6%

Interest earned

at 4 12 %

T

T

y = 0.06x + 0.0451100,000 - x2

= 0.06x + 4500 - 0.045x

Distributive property

= 4500 + 0.015x

Simplify

So the model we want is the linear equation y = 4500 + 0.015x.

(b) Since Mary’s total interest is \$5025, we replace y by 5025 in the model

and solve the resulting one-variable linear equation.

y = 4500 + 0.015x

5025 = 4500 + 0.015x

525 = 0.015x

x =

525

0.015

x = 35,000

Model

Replace y by 5025

Subtract 4500

Divide by 0.015 and switch sides

Calculator

So Mary invested \$35,000 at 6% and the remaining \$65,000 at 4 12 % .

NOW TRY EXERCISE 27

Many real-world problems involve mixing different types of substances. For example, construction workers may mix cement, gravel, and sand; fruit juice from concentrate may involve mixing different types of juices. Problems involving mixtures

and concentrations make use of the fact that if an amount x of a substance is dissolved

in a solution with volume V, then the concentration C of the substance is given by

C=

x

V

So if 10 grams of sugar are dissolved in 5 liters of water, then the sugar concentration is C = 10>5 = 2 g/L.

e x a m p l e 3 Constructing and Using a Model

(Mixtures and Concentration)

A manufacturer of soft drinks advertises its orange soda as “naturally flavored,” although the soda contains only 5% orange juice. A new federal regulation stipulates

that to be called “natural,” a drink must contain at least 10% fruit juice. The manufacturer has a 900-gallon vat of soda and decides to add pure orange juice to the vat.

(a) Construct a model that gives the fraction of the mixture that is pure orange juice.

(b) How much pure orange juice must be added for the mixture to satisfy the

10% rule?

Solution

(a) Choose the variable. The variable in this problem is the amount of pure

orange juice added to the vat. So let

x = the amount 1in gallons2 of pure orange juice added

SECTION 2.6

Linear Equations: Getting Information from a Model

205

Translate words to algebra. Let’s translate all the information given in the

problem into the language of algebra. First note that to begin with, 5% of the

900 gallons in the vat is orange juice, so the amount of orange juice in the vat

is 10.052900 = 45 gallons.

In Words

In Algebra

Amount of the mixture

Amount of orange juice in the mixture

x

900 + x

45 + x

Set up the model. We are now ready to set up the model. To find the

fraction of the mixture that is orange juice, we divide the amount of orange

juice in the mixture by the amount of the mixture.

y =

45 + x

900 + x

d Amount of orange

juice in the mixture

d Amount of the mixture

So the model we want is the equation y = 145 + x 2>1900 + x2 .

(b) We want the mixture to have 10% orange juice. This means that we want the

fraction of the mixture that is orange juice to be 0.10. So we replace y by 0.10

in our model and solve the resulting one-variable linear equation for x.

0.10 =

45 + x

900 + x

0.101900 + x 2 = 45 + x

90 + 0.10x = 45 + x

45 = 0.90x

x =

45

0.90

x = 50

Replace y by 0.10

Cross multiply

Distributive Property

Subtract 45, subtract 0.10x

Divide by 0.90 and switch sides

Calculator

So the manufacturer should add 50 gallons of pure orange juice to the soda.

NOW TRY EXERCISE 29

e x a m p l e 4 Constructing and Using a Model (Geometry)

Al paints with water colors on a sheet of paper that is 20 inches wide by 15 inches

high. He then places this sheet on a mat so that a uniformly wide strip of the mat

shows all around the picture.

(a) Construct a model that gives the perimeter of the mat.

(b) If the perimeter of the mat is 102 inches, how wide is the strip showing around

the picture?

Solution

(a) Choose the variable.

So let

The variable in this problem is the width of the strip.

x = the width of the strip

206

CHAPTER 2

Linear Functions and Models

Translate words to algebra. Figure 2 helps us to translate information

given in the problem into the language of algebra.

In Words

In Algebra

Width of mat

Length of mat

20 + 2x

15 + 2x

15 in.

x

Set up the model. We are now ready to set up the model. To find the

perimeter of the mat, we add twice the width and twice the length.

20 in.

Width

Length

u

u

figure 2

y = 2120 + 2x2 + 2115 + 2x 2

= 40 + 4x + 30 + 4x

Distributive Property

= 70 + 8x

Simplify

So the model we want is the equation y = 70 + 8x.

(b) Since the perimeter is 102 inches, we replace y by 102 in our model and solve

the resulting one-variable equation for x.

y = 70 + 8x

102 = 70 + 8x

8x = 32

x =

Model

Replace y by 102

Subtract 70, switch sides

32

8

Divide by 8

x =4

Calculate

So if the perimeter is 102 inches, the width of the strip is 4 inches.

NOW TRY EXERCISE 31

Check your knowledge of solving linear equations by doing the following problems. You can review these topics in Algebra Toolkit C.1 on page T47.

1. Determine whether each given value is a solution of the equation.

(a) 4x + 7 = 9x - 3;

2, 3

(b)

x+6

(c)

= 5; 1, - 1

x+2

4x - 6

= 2x - 6;

3

0, 6

2. Solve the given equation.

(a) 2x + 7 = 31

(b) 12 1x - 8 2 = 1

3

z

(d) + 3 =

z + 7 (e) 3.2x + 1.4 = 10.9

5

10

(f) - 21x - 1 2 = 512x + 32 - 5x

(c) 31t - 82 = 211 - 5t2

3. Solve the given equation. Check that the solution satisfies the equation.

x

=3

3x - 8

1

1

x +3

(d)

+

=

x + 1 x - 2 x2 - x - 2

(a)

2

3

=

t +6 t-1

5

x +1

(e) 2 +

=

x -4 x -4

(b)

(c)

2x - 2 4

=

x+2

5

SECTION 2.6

Linear Equations: Getting Information from a Model

207

2.6 Exercises

CONCEPTS

Fundamentals

1. (a) The equation y = 8 + 5x is a ______-variable equation.

(b) The equation 3 = 8 + 5x is a ______-variable equation.

2. (a) To find the x-value when y is 6 in the equation y - 2 = 3x + 1, we solve the onevariable equation ________.

(b) The solution of the one-variable equation in part (a) is ________.

3. Suppose the two-variable equation y = 500 + 10x models the cost y of manufacturing

x flash drives. If the company spends \$4000 on manufacturing flash drives, then to find

how many drives were manufactured, we solve the equation ________.

4. Alice invests \$1000 in two certificates of deposit; the interest on the first certificate is

4%, and the interest on the second is 5%. Let’s denote the amount she invests in the first

certificate by x. Then the amount she invests in the second certificate is __________.

The interest she receives on the first certificate is __________, and the interest she

receives on the second certificate is _________. So a function that models the total

interest she receives from both certificates is y = _________.

5–6

True or false?

5. Bill is draining his aquarium. The equation y = 50 - 2x models the volume y of water

remaining in the tank after x minutes. To find how long it takes for the tank to empty,

we replace y by 0 in the model and solve for x.

6. The equation y = - 15 + 3x models the profit y a street vendor makes from selling x

sandwiches. To find the number of sandwiches he must sell to make a profit of \$45, we

replace x by 45 in the model and solve for y.

SKILLS

7–10

Find the value of x that satisfies the given equation when y is 4.

7. y = 24 + 5x

8. y = 6 - 4x

9. 2x - 3y = 8

11–16

10. - 3y = 8 + 2x

The given equation models the relationship between the quantities x and y. Find

the value of x for the given value of y.

11. y = 5 + 2x; 25

13. y = 0.05x + 0.0611000 - x 2 ;

15. y =

17–20

10 + x

;

50 + x

0.5

12. y = 500 - 0.25x;

57.50

100

14. y = 0.025x + 0.035150,000 - x 2 ;

16. y =

1350

5000 + 2x

; 8

250 + x

A quantity Q is related to a quantity R by the given equation. Find the value of Q

for the given value of R.

17.

12

= 21Q - 32 ; 5

R +1

18.

5

= 412Q + 12 ; 3

R -2

19.

Q-5

2

=

;

R 2Q - 1

20.

5 - 2Q

6

=

; -5

R - 1 3Q - 4

-2

208

CHAPTER 2

CONTEXTS

Linear Functions and Models

21. Weather Balloon

A weather balloon is being filled. The linear equation

V = 2 + 0.05t

models the volume V (in cubic feet) of hydrogen in the balloon at any time t (in

seconds). (See Exercise 55 in Section 2.2.) How many minutes will it take until the

balloon contains 55 ft 3 of hydrogen?

22. Filling a Pond A large koi pond is being filled. The linear equation

y = 300 + 10x

models the number y of gallons of water in the pool after x minutes. (See Exercise 56 in

Section 2.2.) If the pond has a capacity of 4000 gallons, how many minutes will it take

for the pond to be filled? Answer this question in two ways:

(a) Graphically (by graphing the equation and estimating the time from the graph)

(b) Algebraically (by solving an appropriate equation)

23. Landfill A county landfill has a maximum capacity of about 131,000,000 tons of

trash. The amount in the landfill on a given day since 1996 is modeled by the function

T 1x 2 = 32,400 + 4x

Here x is the number of days since January 1, 1996, and T 1x2 is measured in thousands

of tons. (See Exercise 53 in Section 2.2.) After how many days will the landfill reach

maximum capacity? Answer this question in two ways:

(a) Graphically (by graphing the equation and estimating the time from the graph)

(b) Algebraically (by solving an appropriate equation)

24. Air Traffic Controller An aircraft is approaching an international airport. Using radar,

an air traffic controller determines that the linear equation

y = - 4x + 45

models the distance (measured in miles) of the approaching aircraft from the radar

tower x minutes since the radar identified the aircraft. (See Exercise 57 in Section 2.3.)

How many minutes will it take for the aircraft to reach the radar tower?

25. Crickets and Temperature Biologists have observed that the chirping rate of a

certain species of cricket is modeled by the linear equation

t =

5

24 n

+ 45

where t is the temperature (in degrees Fahrenheit) and n is the number of chirps per

minute. (See Exercise 64 in Section 2.3.) If the temperature is 80°F, estimate the

cricket’s chirping rate (in chirps per minute).

26. Manufacturing Cost The manager of a furniture factory finds that the cost C (in

dollars) to manufacture x chairs is modeled by the linear equation

C = 13x + 900

How many chairs are manufactured if the cost is \$3500?

27. Investment Lili invests \$12,000 in two one-year certificates of deposit. One

certificate pays 4%, and the other pays 4 12 % simple interest annually.

(a) Construct a model for the total interest Lili earns in one year on her investments.

(Let x represent the amount invested at 4%.)

(b) If Lili’s total interest is \$526.00, how much money did she invest in each

certificate?

28. Investment Ronelio invests \$20,000 in two one-year certificates of deposit. One

certificate pays 3%, and the other pays 3 34 % simple interest annually.

SECTION 2.6

Linear Equations: Getting Information from a Model

209

(a) Construct a model for the total interest Ronelio earns in one year on his

investments. (Let x represent the amount invested at 3%.)

(b) If Ronelio’s total interest is \$697.50, how much money did he invest in each

certificate?

29. Mixture A jeweler has five rings, each weighing 18 grams, made of an alloy of 10%

silver and 90% gold. He decides to melt down the rings and add enough silver to reduce

the gold content to 75%.

(a) Construct a model that gives the fraction of the new alloy that is pure gold. (Let x

represent the number of grams of silver added.)

(b) How much pure silver must be added for the mixture to have a gold content of 75%?

30. Mixture Monique has a pot that contains 6 liters of brine (salt water) at a

concentration of 120 g/L. She needs to boil some of the water off to increase the

concentration of salt.

(a) Construct a model that gives the concentration of the brine after boiling it. (Let x

represent the number of liters of water that is boiled off.)

(b) How much water must be boiled off to increase the concentration of the brine to

200 g/L?

31. Geometry A graphic artist needs to construct a design that uses a rectangle whose

length is 5 cm longer than its width x.

(a) Construct a model that gives the perimeter of the rectangle.

(b) If the perimeter of the rectangle is 26 cm, what are the dimensions of the rectangle?

32. Geometry An architect is designing a building whose footprint has the shape

shown below.

(a) Construct a model that gives the total area of the footprint of the building.

(b) Find x such that the area of the building is 144 square meters.

x

10 m

6m

x

Ô

x

33. Law of the Lever The figure at left shows a lever system, similar to a seesaw that you

might find in a children’s playground. For the system to balance the product of the

weight and its distance from the fulcrum must be the same on each side; that is

w1x1 ϭ w2x2

5 ft

This equation is called the Law of the Lever and was first discovered by Archimedes

(see page 568).

A mother and her son are playing on a seesaw. The boy is at one end, 8 ft from the

fulcrum. If the boy weights 100 lb and the mother weigh 125 lb, at what distance from

the fulcrum should the woman sit so that the seesaw is balanced?

34. Law of the Lever A 30 ft plank rests on top of a flat roofed building, with 5 ft of the

plank projecting over the edge, as shown in the figure. A 240 lb worker sits on one end

of the plank. What is the largest weight that can be hung on the projecting end of the

plank if it is to remain in balance? (Use the Law of the Lever as stated in Exercise 33.)

210

2

CHAPTER 2

Linear Functions and Models

2.7 Linear Equations: Where Lines Meet

Where Lines Meet

Modeling Supply and Demand

IN THIS SECTION… we learn how to find where the graphs of two linear functions

intersect. To find intersection points algebraically, we need to solve equations like the ones

we solved in the preceding section.

GET READY… by reviewing Section 1.7 on how to find graphically where the graphs of

two functions meet.

In many situations involving two linear functions we need to find the point where the

graphs of these functions intersect. In Section 1.7 we used the graphs of functions to

find their intersection points. In this section we will learn to use algebraic techniques

to find the intersection point of two linear functions.

■ Where Lines Meet

A well-known story from Aesop’s fables is about a race between a tortoise and a

hare. The hare is so confident of winning that he decides to take a long nap at the beginning of the race. When he wakes up, he sees that the tortoise is very close to the

finish line; he realizes that he can’t catch up.

The hare’s confidence is justified; hares can sprint at speeds of up to 30 mi/h,

whereas tortoises can barely manage 0.20 mi/h. But let’s say that in this 2-mile race,

the tortoise tries really hard and keeps up a 0.8 mi/h pace; the complacent hare runs

at a leisurely 4 mi/h after a two and a quarter hour nap. Under these conditions, who

will win the race? Let’s see.

The tortoise starts the race at time zero, so the distance y he reaches in the race

course at time t is modeled by the equation y = 0 + 0.8t or

The hare is much faster than the

tortoise, but will he win the race?

y = 0.8t

Tortoise equation

Let’s find the linear equation that models the distance the hare reaches at time t. The

hare’s distance y is 0 when t is 2.25, so the point (2.25, 0) is on the graph of the desired equation. Using the point-slope formula, we get y - 0 = 41t - 2.252 , or

y = - 9 + 4t

Hare equation

for t Ú 2.25. The graph in Figure 1 shows that the tortoise reaches the two-mile finish line before the hare does.

y

3

1

f i g u r e 1 The hare and

tortoise race

0

Finish line

e

ois

rt

To

1

e

2

Hare overtakes

tortoise

Har

2

2

3

t

SECTION 2.7

Linear Equations: Where Lines Meet

211

If the race is to continue beyond the finish line, when does the hare overtake the

tortoise? We can answer this question graphically (as in Section 1.7) or algebraically:

Graphically: From Figure 1 we estimate that the intersection point of the two

lines is approximately (2.8, 2.2). This point tells us that the hare overtakes the

tortoise about 2.8 hours into the race.

Algebraically: The hare and the tortoise have reached the same point in the

race course when the y-values (the distance traveled) in each equation are

equal. So we equate the y-values from each equation:

The accuracy of the graphical

method depends on how precisely

we can draw the graph. The

algebraic method always gives us

0.8t = - 9 + 4t

Solving for t, we get t = 2.8125. So the rabbit overtakes the hare 2.8125 hours

into the race.

In general, the graphs of the functions f and g intersect at those values of x for

which f 1x2 = g1x2 . In the special case in which the functions are linear, we have the

following.

Intersection Points of Linear Functions

To find where the graphs of the linear functions

y = b1 + m1x

and

y = b2 + m2x

intersect, we solve for x in the equation

b1 + m1x = b2 + m2x

e x a m p l e 1 Finding Where Two Lines Meet

Let f 1x 2 = - 8 + 5x and g1x2 = 2 + 3x. Find the value of x where the graphs of f

and g intersect, and find the point of intersection.

Solution 1 Graphical

y

10

g

0

2

f

x

We graph f and g in Figure 2. From the graph we see that the x-value of the intersection point is about x ϭ 5. Also from the graph we see that the y-value of the intersection point is more than 15, but it is difficult to see its exact value from the graph. So

from the graph we can estimate that the intersection point is (5, 15). (Note that this

answer is only an approximation and depends on how precisely we can draw the

graph.)

Solution 2 Algebraic

f i g u r e 2 Graphs of f and g

We solve the equation

- 8 + 5x = 2 + 3x

Set functions equal

5x = 10 + 3x

2x = 10

Subtract 3x

x=5

Divide by 2

So f 1x2 = g 1x2 when x ϭ 5. This means that the value of x where the graphs intersect is 5.

CHAPTER 2

Linear Functions and Models

Let’s find the common value of f and g when x ϭ 5:

f 152 = - 8 + 5152 = - 8 + 25 = 17

g1x2 = 2 + 5132 = 17

So the point (5, 17) is on both graphs, thus the graphs intersect at (5, 17).

IN CONTEXT ➤

NOW TRY EXERCISE 5

With rising fuel costs, there is an increasing interest in hybrid-electric vehicles,

which consume a lot less fuel than gas-powered vehicles do. Also, several manufacturers are now introducing fully electric cars, including some sporty models. The

Tesla fully electric sports car outperforms many gas-powered sports cars. The Tesla

can accelerate from 0 to 60 mi/h in 3.9 seconds. That matches the Lamborghini

Diablo’s acceleration. So now it is possible to have a fully electric car of your

dreams! Moreover, electric motors have few moving parts, so they require far less

maintenance (for example, the Tesla’s electric motor has only twelve moving parts).

Jonathan Larsen/Shutterstock.com 2009

212

Plug-in hybrid-electric car

Fully electric sports car

e x a m p l e 2 Cost Comparison of Gas-Powered and Hybrid-Electric Cars

Kevin needs to buy a new car. He compares the cost of owning two cars he likes:

Car A: An \$18,000 gas-powered car that gets 20 mi/gal

Car B: A \$25,000 hybrid-electric car that gets 48 mi/gal

For this comparison he assumes that the price of gas is \$4.50 per gallon.

(a) Find a linear equation that models the cost of purchasing Car A and driving it

x miles.

(b) Find a linear equation that models the cost of purchasing Car B and driving it

x miles.

(c) Find the break-even point for Kevin’s cost comparison. That is, find the

number of miles he needs to drive so that the cost of owning Car A is the same

as the cost of owning Car B.

Solution

(a) Since the gas-powered car gets 20 miles per gallon and the price of gas is

assumed to be \$4.50, the cost of driving x miles is 4.501x>202 = 0.225x. Since

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