8.3
Systems of Linear Equations:
Solving by Substitution
8.3
OBJECTIVES
1. Solve systems using the substitution method
2. Solve applications of systems of equations
In Sections 8.1 and 8.2, we looked at graphing and addition as methods of solving linear
systems. A third method is called solution by substitution.
Example 1
Solving a System by Substitution
Solve by substitution.
x y 12
(1)
y 3x
(2)
Notice that equation (2) says that y and 3x name the same quantity. So we may substitute 3x
for y in equation (1). We then have
Replace y with 3x in
equation (1).
NOTE The resulting equation
contains only the variable x, so
substitution is just another way
of eliminating one of the
variables from our system.
x 3x 12
4x 12
x3
We can now substitute 3 for x in equation (1) to find the corresponding y coordinate of the
solution.
3 y 12
y9
NOTE The solution for a
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system is written as an ordered
pair.
So (3, 9) is the solution.
This last step is identical to the one you saw in Section 8.2. As before, you can substitute the known coordinate value back into either of the original equations to find the value
of the remaining variable. The check is also identical.
CHECK YOURSELF 1
Solve by substitution.
xy9
y 4x
The same technique can be readily used any time one of the equations is already solved
for x or for y, as Example 2 illustrates.
657
658
CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
Example 2
Solving a System by Substitution
Solve by substitution.
2x 3y 3
(1)
y 2x 7
(2)
Because equation (2) tells us that y is 2x 7, we can replace y with 2x 7 in equation (1).
This gives
y
from the equation, and we can
proceed to solve for x.
2x 3(2x 7) 3
2x 6x 21 3
8x 24
x3
We now know that 3 is the x coordinate for the solution. So substituting 3 for x in equation (2), we have
y237
67
1
And (3, 1) is the solution. Once again you should verify this result by letting x 3 and
y 1 in the original system.
CHECK YOURSELF 2
Solve by substitution.
2x 3y 6
x 4y 2
As we have seen, the substitution method works very well when one of the given equations is already solved for x or for y. It is also useful if you can readily solve for x or for y
in one of the equations.
Example 3
Solving a System by Substitution
Solve by substitution.
x 2y 5
(1)
3x y 8
(2)
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NOTE Now y is eliminated
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION
SECTION 8.3
659
Neither equation is solved for a variable. That is easily handled in this case. Solving for x
in equation (1), we have
x 2y 5
been solved for y with the
result substituted into
equation (1).
Now substitute 2y 5 for x in equation (2).
x
NOTE Equation (2) could have
3(2y 5) y 8
6y 15 y 8
7y 7
y 1
Substituting 1 for y in equation (2) yields
3x (1) 8
3x 9
x3
So (3, 1) is the solution. You should check this result by substituting 3 for x and 1 for y
in the equations of the original system.
CHECK YOURSELF 3
Solve by substitution.
3x y 5
x 4y 6
Inconsistent systems and dependent systems will show up in a fashion similar to that
which we saw in Section 8.2. Example 4 illustrates this approach.
Example 4
Solving an Inconsistent or Dependent System
Solve the following systems by substitution.
(a) 4x 2y 6
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y 2x 3
From equation (2) we can substitute 2x 3 for y in equation (1).
4x 2(2x 3) 6
NOTE Don’t forget to change
both signs in the parentheses.
4x 4x 6 6
66
Both variables have been eliminated,
and we have the true statement
6 6.
(1)
(2)
CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
Recall from the last section that a true statement tells us that the lines coincide. We call this
system dependent. There are an infinite number of solutions.
(b) 3x 6y 9
(3)
x 2y 2
(4)
Substitute 2y 2 for x in equation (3).
3(2y 2) 6y 9
6y 6 6y 9
69
This time we have
a false statement.
This means that the system is inconsistent and that the graphs of the two equations are
parallel lines. There is no solution.
CHECK YOURSELF 4
Indicate whether the systems are inconsistent (no solution) or dependent (an
infinite number of solutions).
(a) 5x 15y 10
x 3y 1
(b) 12x 4y 8
y 3x 2
The following summarizes our work in this section.
Step by Step: To Solve a System of Linear Equations
by Substitution
Step 1
Step 2
Step 3
Step 4
Step 5
Solve one of the given equations for x or y. If this is already done, go on
to step 2.
Substitute this expression for x or for y into the other equation.
Solve the resulting equation for the remaining variable.
Substitute the known value into either of the original equations to find
the value of the second variable.
Check your solution in both of the original equations.
You have now seen three different ways to solve systems of linear equations: by graphing,
adding, and substitution. The natural question is, Which method should I use in a given
situation?
Graphing is the least exact of the methods, and solutions may have to be estimated.
The algebraic methods—addition and substitution—give exact solutions, and both
will work for any system of linear equations. In fact, you may have noticed that several
examples in this section could just as easily have been solved by adding (Example 3,
for instance).
The choice of which algebraic method (substitution or addition) to use is yours and
depends largely on the given system. Here are some guidelines designed to help you choose
an appropriate method for solving a linear system.
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660
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION
SECTION 8.3
661
Rules and Properties: Choosing an Appropriate Method for
Solving a System
1. If one of the equations is already solved for x (or for y), then substitution is
the preferred method.
2. If the coefficients of x (or of y) are the same, or opposites, in the two
equations, then addition is the preferred method.
3. If solving for x (or for y) in either of the given equations will result in
fractional coefficients, then addition is the preferred method.
Example 5
Choosing an Appropriate Method for Solving a System
Select the most appropriate method for solving each of the following systems.
(a) 5x 3y 9
2x 7y 8
Addition is the most appropriate method because solving for a variable will result in
fractional coefficients.
(b) 7x 26 8
x 3y 5
Substitution is the most appropriate method because the second equation is already solved
for x.
(c) 8x 9y 11
4x 9y 15
Addition is the most appropriate method because the coefficients of y are opposites.
CHECK YOURSELF 5
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Select the most appropriate method for solving each of the following systems.
(a) 2x 5y 3
8x 5y 13
(b) 4x 3y 2
y 3x 4
(c) 3x 5y 2
x 3y 2
(d) 5x 2y 19
4x 6y 38
Number problems, such as those presented in Chapter 2, are sometimes more easily
solved by the methods presented in this section. Example 6 illustrates this approach.
Example 6
Solving a Number Problem by Substitution
The sum of two numbers is 25. If the second number is 5 less than twice the first number,
what are the two numbers?
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CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
NOTE
Step 1 You want to find the two unknown numbers.
1. What do you want to find?
2. Assign variables. This time
we use two letters, x and y.
Step 2 Let x the first number and y the second number.
3. Write equations for the
solution. Here two equations
are needed because we have
introduced two variables.
Step 3
x y 25
The sum
is 25.
y 2x 5
The second
number
4. Solve the system of
equations.
is 5 less than
twice the first.
Step 4
x y 25
(1)
y 2x 5
NOTE We use the substitution
Substitute 2x 5 for y in equation (1).
method because equation (2) is
already solved for y.
x (2x 5) 25
(2)
3x 5 25
x 10
From equation (1),
10 y 25
y 15
The two numbers are 10 and 15.
Step 5 The sum of the numbers is 25. The second number, 15, is 5 less than twice the
first number, 10. The solution checks.
CHECK YOURSELF 6
The sum of two numbers is 28. The second number is 4 more than twice the first
number. What are the numbers?
Sketches are always helpful in solving applications from geometry. Let’s look at such an
example.
Example 7
Solving an Application from Geometry
The length of a rectangle is 3 meters (m) more than twice its width. If the perimeter of the
rectangle is 42 m, find the dimensions of the rectangle.
Step 1 You want to find the dimensions (length and width) of the rectangle.
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5. Check the result.
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION
NOTE We used x and y as our
two variables in the previous
examples. Use whatever letters
you want. The process is the
same, and sometimes it helps
you remember what letter
stands for what. Here L length and W width.
SECTION 8.3
663
Step 2 Let L be the length of the rectangle and W the width. Now draw a sketch of the
problem.
L
W
W
L
Step 3 Write the equations for the solution.
L 2W 3
3 more than twice
the width
2L 2W 42
The perimeter
Step 4 Solve the system.
L 2W 3
2L 2W 42
(2)
NOTE Substitution is used
From equation (1) we can substitute 2W 3 for L in equation (2).
because one equation is already
solved for a variable.
2(2W 3) 2W 42
4W 6 2W 42
6W 36
W6
Replace W with 6 in equation (1) to find L.
L263
12 3
15
The length is 15 m, the width is 6 m.
6m
15 m
Step 5 Check these results. The perimeter is 2L 2W, which should give us 42 m.
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(1)
2(15) 2(6) 42
30 12 42
CHECK YOURSELF 7
The length of each of the two equal legs of an isosceles triangle is 5 in. less than the
length of the base. If the perimeter of the triangle is 50 in., find the lengths of the
legs and the base.
CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
CHECK YOURSELF ANSWERS
1.
4.
5.
6.
(3, 12)
2. (6, 2)
3. (2, 1)
(a) Inconsistent system; (b) Dependent system
(a) Addition; (b) Substitution; (c) Substitution; (d) Addition
The numbers are 8 and 20.
7. The legs have length 15 in.; the base is 20 in.
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664
Name
8.3
Exercises
Section
Date
Solve each of the following systems by substitution.
1. x y 10
y 4x
2. x y 4
x 3y
ANSWERS
1.
2.
3. 2x y 10
x 2y
4. x 3y 10
3x y
3.
4.
5. 3x 2y 12
y 3x
6. 4x 3y 24
y 4x
5.
6.
7. x y 5
yx3
8. x y 9
xy3
7.
8.
9. x y 4
x 2y 2
10. x y 7
y 2x 12
9.
10.
11.
11. 2x y 7
y x 8
12. 3x y 15
xy7
12.
13.
13. 2x 5y 10
xy8
14. 4x 3y 0
14.
yx1
15.
15. 3x 4y 9
y 3x 1
16. 5x 2y 5
y 5x 3
16.
17.
18.
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17. 3x 18y 4
x 6y 2
18. 4x 5y 6
y 2x 10
19.
20.
19. 5x 3y 6
y 3x 6
20. 8x 4y 16
y 2x 4
21.
22.
21. 8x 5y 16
y 4x 5
22. 6x 5y 27
x 5y 2
665
ANSWERS
23.
23. x 3y 7
24. 2x y 4
x y3
24.
25.
25.
26.
27.
27.
x y 5
6x 3y 9
2x y 3
26. 5x 6y 21
x 7y 3
2x 5y 15
28. 4x 12y x 2y 5
5
x 3y 1
28.
29. 4x 3y 11
29.
5x y 11
30. 5x 4y 5
4x y 7
30.
31.
Solve each of the following systems by using either addition or substitution. If a unique
solution does not exist, state whether the system is dependent or inconsistent.
32.
33.
31. 2x 3y 6
x 3y 6
34.
35.
33.
36.
2x y 1
2x 3y 5
35. 6x 2y 4
37.
y 3x 2
32. 7x 3y 31
y 2x 9
34.
x 3y 12
2x 3y 6
36. 3x 2y 15
x 5y 5
38.
40.
x 2y 2
3x 2y 12
y 5x 3
39. 2x 3y 14
40. 2x 3y 1
41. 4x 2y 0
42. 4x 3y 4x 5y 5
41.
42.
38. 10x 2y 7
43.
x
5x 3y 16
11
2
3
y
2
3
2
44.
Solve each system.
43.
666
1
1
x y 5
3
2
x
y
2
4
5
44.
5x
9
y 2
10
3x
5y
2
4
6
3
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37.
39.
ANSWERS
45. 0.4x 0.2y 0.6
2.5x 0.3y 4.7
46. 0.4x 0.1y 5
6.4x 0.4y 60
45.
46.
47.
Solve each of the following problems. Be sure to show the equation used for the solution.
48.
47. Number problem. The sum of two numbers is 100. The second is three times the
first. Find the two numbers.
49.
50.
48. Number problem. The sum of two numbers is 70. The second is 10 more than
51.
3 times the first. Find the numbers.
52.
49. Number problem. The sum of two numbers is 56. The second is 4 less than twice
53.
the first. What are the two numbers?
54.
50. Number problem. The difference of two numbers is 4. The larger is 8 less than
twice the smaller. What are the two numbers?
55.
56.
51. Number problem. The difference of two numbers is 22. The larger is 2 more than
3 times the smaller. Find the two numbers.
52. Number problem. One number is 18 more than another, and the sum of the smaller
number and twice the larger number is 45. Find the two numbers.
53. Number problem. One number is 5 times another. The larger number is 9 more
than twice the smaller. Find the two numbers.
54. Package weight. Two packages together weigh 32 kilograms (kg). The smaller
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package weighs 6 kg less than the larger. How much does each package weigh?
55. Appliance costs. A washer-dryer combination costs $1200. If the washer costs
$220 more than the dryer, what does each appliance cost separately?
56. Voting trends. In a town election, the winning candidate had 220 more votes
than the loser. If 810 votes were cast in all, how many votes did each candidate
receive?
667
ANSWERS
57.
57. Cost of furniture. An office desk and chair together cost $850. If the desk cost $50
less than twice as much as the chair, what did each cost?
58.
59.
60.
a.
b.
58. Dimensions of a rectangle. The length of a rectangle is 2 inches (in.) more than
twice its width. If the perimeter of the rectangle is 34 in., find the dimensions of the
rectangle.
59. Perimeter. The perimeter of an isosceles triangle is 37 in. The lengths of the two
equal legs are 6 in. less than 3 times the length of the base. Find the lengths of the
three sides.
60. You have a part-time job writing the Consumer Concerns column for your local
newspaper. Your topic for this week is clothes dryers, and you are planning to
compare the Helpmate and the Whirlgarb dryers, both readily available in stores in
your area. The information you have is that the Helpmate dryer is listed at $520, and
it costs 22.5¢ to dry an average size load at the utility rates in your city. The
Whirlgarb dryer is listed at $735, and it costs 15.8¢ to run for each normal load. The
maintenance costs for both dryers are about the same. Working with a partner, write a
short article giving your readers helpful advice about these appliances. What should
they consider when buying one of these clothes dryers?
Getting Ready for Section 8.4 [Section 2.7]
Graph the solution sets for the following linear inequalities.
(b) 2x y 6
y
y
x
668
x
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(a) x y 8
ANSWERS
(c) 3x 4y 12
c.
(d) y 2x
d.
y
y
e.
f.
x
x
(e) y 3
(f) x 5
y
y
x
x
Answers
1. (2, 8)
13. (10, 2)
23. (4, 1)
3. (4, 2)
15.
3, 2
1
4
7. (4, 1)
17. No solution
33. (2, 3)
2, 3
5
41.
2, 3
3
43. (0, 10)
11. (5, 3)
9. (10, 6)
19. (3, 3)
27. (10, 1)
35. Dependent system
49. 20, 36
51. 32, 10
57. Desk $550, chair $300
37.
45. (2, 1)
21.
4, 2
3
29. (2, 1)
5, 2
3
47. 25, 75
53. 3, 15
55. Washer $710, dryer $490
59. 7 in., 15 in., 15 in.
a. x y 8
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3, 4
25. Infinite number of solutions
31. (0, 2)
39.
5.
b. 2x y 6
y
y
x
x
669
c. 3x 4y 12
d. y 2x
y
y
x
e. y 3
x
f. x 5
y
y
x
© 2001 McGraw-Hill Companies
x
670