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Systems of Linear Equations and Inequalities
44. Flat tax proposals. Representative Schneider has proposed a
flat income tax of 15% on earnings in excess of $10,000.
Under his proposal the tax T for a person earning E dollars is
given by T 0.15(E 10,000). Representative Humphries
has proposed that the income tax should be 20% on earnings
in excess of $20,000, or T 0.20(E 20,000). Graph both
linear equations on the same coordinate system. For what
earnings would you pay the same amount of income tax under
either plan? Under which plan does a rich person pay less
income tax? $50,000, Schneider’s plan
46. Cooperative learning. Working in groups, write an independent system of two linear equations whose solution is
(3, 5). Each group should then give its system to another
group to solve.
x y 8, x y 2
47. Cooperative learning. Working in groups, write an inconsistent system of linear equations such that (2, 3) satisfies
one equation and (1, 4) satisfies the other. Each group
should then give its system to another group to solve.
x y 1, x y 5
G R A P H I N G C ALC U L ATO R
EXERCISES
Solve each system by graphing each pair of equations on a
graphing calculator and using the calculator to estimate the
point of intersection. Give the coordinates of the intersection to
the nearest tenth.
GET TING MORE INVOLVED
48. y 2.5x 6.2
y 1.3x 8.1
(3.8, 3.2)
49. y 305x 200
y 201x 999
(2.4, 522.7)
45. Discussion. If both (1, 3) and (2, 7) satisfy a system of
two linear equations, then what can you say about the
system? It is a dependent system.
50. 2.2x 3.1y 3.4
5.4x 6.2y 7.3
(1.4, 0.1)
51. 34x 277y 1
402x 306y 12,000
(27.3, 3.3)
7.2
In this
section
THE SUBSTITUTION METHOD
Solving a system by graphing is certainly limited by the accuracy of the graph. If the
lines intersect at a point whose coordinates are not integers, then it is difficult to
identify the solution from a graph. In this section we introduce a method for solving
systems of linear equations in two variables that does not depend on a graph and is
totally accurate.
●
Solving a System of Linear
Equations by Substitution
●
Inconsistent and
Dependent Systems
Solving a System of Linear Equations by Substitution
●
Applications
The next example shows how to solve a system without graphing. The method is
called substitution.
E X A M P L E
1
Solving a system by substitution
Solve:
2x 3y 9
y 4x 8
Solution
First solve the second equation for y:
y 4x 8
y 4x 8
7.2
The Substitution Method
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377
Now substitute 4x 8 for y in the first equation:
calculator
2x 3y 9
2x 3(4x 8) 9 Substitute 4x 8 for y.
2x 12x 24 9 Simplify.
10x 24 9
10x 15
15
x 10
3
2
close-up
To check Example 1, graph
y1 (2x 9)3
and
y2 4x 8.
Use the intersect feature of
your calculator to find the
point of intersection.
10
3
2
Use the value x in y 4x 8 to find y :
–10
3
y 4 8
2
2
10
–10
2, 2 satisfies both of the original equations. The solution to the
system is 3, 2.
■
2
Check that
E X A M P L E
2
3
Solving a system by substitution
Solve:
3x 4y 5
xy1
Solution
Because the second equation is already solved for x in terms of y, we can substitute y 1 for x in the first equation:
calculator
3x 4y 5
3( y 1) 4y 5
3y 3 4y 5
7y 3 5
7y 8
8
y 7
close-up
To check Example 2, graph
y1 (5 3x)4
and
y2 x 1.
Use the intersect feature of
your calculator to find the
point of intersection.
10
–10
10
–10
Replace x with y 1.
Simplify.
8
Now use the value y 7 in one of the original equations to find x. The simplest one
to use is x y 1:
8
x 1
7
1
x 7
Check that 1, 8 satisfies both equations. The solution to the system is 1, 8.
7 7
The strategy for solving by substitution is given on the next page.
7 7
■
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Chapter 7
Systems of Linear Equations and Inequalities
Strategy for Solving a System by Substitution
1. Solve one of the equations for one variable in terms of the other.
2. Substitute this value into the other equation to eliminate one of the variables.
3. Solve for the remaining variable.
4. Insert this value into one of the original equations to find the value of the
other variable.
5. Check your solution in both equations.
Inconsistent and Dependent Systems
The following examples illustrate how the inconsistent and dependent cases appear
when we use substitution to solve the system.
E X A M P L E
3
calculator
An inconsistent system
Solve by substitution:
3x 6y 9
x 2y 5
close-up
To check Example 3, graph
y1 (3x 9)6 and y2 (x 5)2. Since the lines appear to be parallel, there is no
solution to the system.
Solution
Use x 2y 5 to replace x in the first equation:
3x 6y 9
3(2y 5) 6y 9
6y 15 6y 9
15 9
10
–10
10
Replace x by 2y 5.
Simplify.
No values for x and y will make 15 equal to 9. So there is no ordered pair that satisfies both equations. This system is inconsistent. It has no solution. The equations are
■
the equations of parallel lines.
–10
E X A M P L E
helpful
4
hint
The purpose of Examples
3 and 4 is to show what happens when you try to solve
an inconsistent or dependent
system by substitution. If we
had first written the equations
in slope-intercept form, we
would have seen that the
lines in Example 3 are parallel
and the lines in Example 4
are the same.
A dependent system
Solve:
2(y x) x y 1
y 3x 1
Solution
Because the second equation is solved for y, we will eliminate the variable y in the
substitution. Substitute y 3x 1 into the first equation:
2(3x 1 x) x (3x 1) 1
2(2x 1) 4x 2
4x 2 4x 2
Any value for x makes the last equation true because both sides are identical. So any
value for x can be used as a solution to the original system as long as we choose
7.2
(7-11)
The Substitution Method
379
y 3x 1. The system is dependent. The two equations are equations for the same
straight line. The solution to the system is the set of all points on that line,
(x, y) y 3x 1.
■
When solving a system by substitution, we can recognize an inconsistent system
or dependent system as follows:
Inconsistent and Dependent Systems
An inconsistent system leads to a false statement.
A dependent system leads to a statement that is always true.
Applications
Many of the problems that we solved in previous chapters had two unknown
quantities, but we wrote only one equation to solve the problem. For problems
with two unknown quantities we can use two variables and a system of equations.
E X A M P L E
helpful
5
hint
In Chapter 2, we would have
done Example 5 with one
variable by letting x represent
the amount invested at 6%
and 25,000 x represent the
amount invested at 8%.
Two investments
Mrs. Robinson invested a total of $25,000 in two investments, one paying 6% and
the other paying 8%. If her total income from these investments was $1790, then
how much money did she invest in each?
Solution
Let x represent the amount invested at 6%, and let y represent the amount invested
at 8%. The following table organizes the given information.
Interest
rate
Amount
invested
Amount
of interest
First investment
6%
x
0.06x
Second investment
8%
y
0.08y
Write one equation describing the total of the investments, and the other equation
describing the total interest:
calculator
x y 25,000
0.06x 0.08y 1790
Total investments
Total interest
To solve the system, we solve the first equation for y:
close-up
You can use a calculator
to check the answers in Example 5:
y 25,000 x
Substitute 25,000 x for y in the second equation:
0.06x 0.08(25,000 x) 1790
0.06x 2000 0.08x 1790
0.02x 2000 1790
0.02x 210
210
x 0.02
10,500
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Chapter 7
Systems of Linear Equations and Inequalities
Let x 10,500 in the equation y 25,000 x to find y:
y 25,000 10,500
14,500
Check these values for x and y in the original problem. Mrs. Robinson invested
$10,500 at 6% and $14,500 at 8%.
■
M A T H
A T
W O R K
First the special handshake for luck, then climbing
in the coupe, strapping the seat belt on as tightly as
possible, eyes locked forward, thinking who is the
person to beat, where are the bumps and curves . . .
concentration! These are some of the thoughts and
rituals Ossie Babson performs as the driver-partner
RACE CAR
of the Babson Brothers Racing Team. The special
DRIVER
handshake is with his brother Dave Babson, who is
5
the crew chief of the team. The car is a -scale model of a 1940 Ford Coupe,
8
specially built for the Legends of Nascar Racing Series.
There are strict rules on the weight distribution, frame height, tire size, and
engine size for these cars, but for best results the car should be set up to lean to the
left and front. The challenge is to meet all these criteria for a successful race. Before
the race, Ossie drives on the track and makes observations and recommendations on
how the car handles going into the turns and what adjustments to make for better
performance under specific track conditions. Dave then supervises the changes to
the car. This process continues until both driver and crew chief are satisfied. Ultimately, a combination of art, how the car feels, and science makes the car perform
to its ultimate capabilities. As a result of their teamwork last year, the Babsons finished in one of the top ten positions in nine races including an impressive second
place finish.
In Exercises 35 and 36 of this section you will see how the Babsons use a system
of equations to determine the proper weight distribution for their car.
WARM-UPS
True or false? Explain your answer.
For Exercises 1–7, use the following systems:
a) y x 7
b) x 2y 1
2x 3y 4
2x 4y 0
1. If we substitute x 7 for y in system (a), we get 2x 3(x 7) 4.
2. The x-coordinate of the solution to system (a) is 5. True
3. The solution to system (a) is (5, 2). True
( )
4. The point 1, 1 satisfies system (b). True
2 4
5. It would be difficult to solve system (b) by graphing. True
6. Either x or y could be eliminated by substitution in system (b).
True
True
7.2 The Substitution Method
WARM-UPS
(7-13)
381
(continued )
7. System (b) is a dependent system. False
8. Solving an inconsistent system by substitution will result in a false
statement. True
9. Solving a dependent system by substitution results in an equation that is
always true. True
10. Any system of two linear equations can be solved by substitution. True
7. 2
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What method is used in this section to solve systems of
equations?
In this section we learned the substitution method.
2. What is wrong with the graphing method for solving
systems?
Graphing is not accurate enough.
3. What is a dependent system?
A dependent system is one in which the equations are
equivalent.
4. What is an inconsistent system?
An inconsistent system is one with no solution.
18. x 2y 1
3x 10y 1
Solve each system by substitution, and identify each system as
independent, dependent, or inconsistent. See Examples 3 and 4.
19. x 2y 2
x 2y 8
20. y 3x 1
, independent
y 2x 4
21. x 4 2y
4y 2x 8
No solution, inconsistent
5. What happens when you try to solve a dependent system by
substitution?
Using substitution on a dependent system results in an
equation that is always true.
22. 21x 35 7y
3x y 5 (x, y) 3x y 5, dependent
23. y 3 2(x 1)
y 2x 3 No solution, inconsistent
6. What happens when you try to solve an inconsistent system
by substitution?
24. y 1 5(x 1)
y 5x 1 No solution, inconsistent
25. 3x 2y 7
Using substitution on an inconsistent system results in a
false equation.
Solve each system by the substitution method. See Examples 1
and 2.
7. y x 3
8. y x 5
2x 3y 11 (2, 5)
x 2y 8 (6, 1)
9. x 2y 4
10. x y 2
2x y 7 (2, 3)
2x y 1 (3, 5)
11. 2x y 5
5x 2y 8
13. x y 0
3x 2y 5
15. x y 1
4x 8y 4
17. 2x 3y 2
4x 9y 1
(2, 9)
(5, 5)
12. 5y x 0
6x y 2
14. x y 6
3x 4y 3
16. x y 2
3x 6y 8
0 2
(3, 3)
3x 2y 7
26. 2x 5y 5
3x 5y 6
1 3
2
27. x 5y 4
x 5y 4y (1, 1), independent
28. 2x y 3x
3x y 2y (x, y) y x, dependent
Write a system of two equations in two unknowns for each problem. Solve each system by substitution. See Example 5.
29. Investing in the future. Mrs. Miller invested $20,000 and
received a total of $1600 in interest. If she invested part of
the money at 10% and the remainder at 5%, then how
much did she invest at each rate?
$12,000 at 10%, $8000 at 5%
30. Stocks and bonds. Mr. Walker invested $30,000 in stocks
and bonds and had a total return of $2880 in one year. If his
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Chapter 7
Systems of Linear Equations and Inequalities
stock investment returned 10% and his bond investment
returned 9%, then how much did he invest in each?
$18,000 at 10%, $12,000 at 9%
32. Tennis court dimensions. The singles court in tennis is
four yards longer than it is wide. If its perimeter is 44 yards,
then what are the length and width?
Length 13 yd, width 9 yd
33. Mowing and shoveling. When Mr. Wilson came back
from his vacation, he paid Frank $50 for mowing his lawn
three times and shoveling his sidewalk two times. During
Mr. Wilson’s vacation last year, Frank earned $45 for mowing the lawn two times and shoveling the sidewalk three
times. How much does Frank make for mowing the lawn
once? How much does Frank make for shoveling the sidewalk once?
Lawn $12, sidewalk $7
34. Burgers and fries. Donna ordered four burgers and one
order of fries at the Hamburger Palace. However, the waiter
put three burgers and two orders of fries in the bag and
charged Donna the correct price for three burgers and two
orders of fries, $3.15. When Donna discovered the mistake,
she went back to complain. She found out that the price for
four burgers and one order of fries is $3.45 and decided to
keep what she had. What is the price of one burger, and what
is the price of one order of fries?
Burger $0.75, fries $0.45
35. Racing rules. According to Nascar rules, no more than
52% of a car’s total weight can be on any pair of tires. For
optimal performance a driver of a 1150-pound car wants
to have 50% of its weight on the left rear and left front
tires and 48% of its weight on the left rear and right front
tires. If the right front weight is determined to be 264
pounds, then what amount of weight should be on the left
rear and left front? Are the Nascar rules satisfied with this
weight distribution?
Left rear 288 pounds, left front 287 pounds, no
36. Weight distribution. A driver of a 1200-pound car wants
to have 50% of the car’s weight on the left front and left
rear tires, 48% on the left rear and right front tires, and
51% on the left rear and right rear tires. How much weight
should be on each of these tires?
Left front 306, left rear 294, right front 282, right rear
318 pounds
37. Price of hamburger. A grocer will supply y pounds of
ground beef per day when the retail price is x dollars per
pound, where y 200x 60. Consumer studies show that
1000
Quantity (pounds/day)
31. Gross Receipts. On July 12, 1998, after 4 weeks in release,
the gross receipts for Mulan exceeded the gross receipts for
The X-Files by $17.3 million (Entertainment Weekly,
www.ew.com). If the total gross receipts for these two
movies was $166.3 million, then what were the gross
receipts for each movie?
Mulan $91.8 million, X-Files $74.5 million
consumer demand for ground beef is y pounds per day,
where y 150x 900. What is the price at which the
supply is equal to the demand, the equilibrium price? See
the figure below.
$2.40 per pound
Point of
equilibrium
800
Supply
y = 200x + 60
600
400
200
0
Demand
y = –150x + 900
1 2 3 4 5 6
Price of ground beef
(dollars/pound)
FIGURE FOR EXERCISE 37
GR APHING C ALCUL ATOR EXERCISE
38. Life expectancy. Since 1950, the life expectancy of a U.S.
male born in year x is modeled by the formula
y 0.165x 256.7,
and the life expectancy of a U.S. female born in year x is
modeled by
y 0.186x 290.6
(National Center for Health Statistics, www.cdc.gov).
a) Find the life expectancy of a U.S. male born in 1975 and
a U.S. female born in 1975.
b) Graph both equations on your graphing calculator for
1950 x 2050.
c) Will U.S. males ever catch up with U.S. females in life
expectancy?
d) Assuming that these equations were valid before 1950,
solve the system to find the year of birth for which U. S.
males and females had the same life expectancy.
a) 69.2 years, 76.8 years
b)
c) No
d) 1614