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
about 1950 minutes.
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
about 1950 minutes
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 orange juice added
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
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■
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 = _________.
Think About It
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
xÔ
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
the exact answer.
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
Add 8
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).
Image by © Car Culture/Corbis
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