Download 7-2 Solving Systems Using Substitution

7-2
Solving Systems Using
Substitution
7-2
1. Plan
Objectives
1
To solve systems using
substitution
Examples
1
2
3
Using Substitution
Using Substitution and the
Distributive Property
Real-World Problem Solving
What You’ll Learn
Check Skills You’ll Need
• To solve systems using
Solve each equation.
substitution
1. m - 6 = 4m + 8 –423
. . . And Why
GO for Help
Lessons 3-3 and 7-1
3. 13 t + 5 = 10 15
2. 4n = 9 - 2n 112
For each system, is the ordered pair a solution of both equations?
To solve problems involving
transportation, as in
Example 3
y = -x + 4
y = x - 6 no
4. (5, 1)
5. (2, 2.4)
4x + 5y = 20
2x + 6y = 10 no
New Vocabulary • substitution method
Math Background
Solving systems using substitution
begins with the concept of evaluating variable expressions. Here a
variable is replaced with its value
in terms of the other variable.
More Math Background: p. 372C
1
UsingSubstitution
Substitution
1 1 Using
Part
You can solve a system of equations by graphing when the solution contains
integers or when you have a graphing calculator. Another method for solving
systems of equations is the substitution method. By replacing one variable with
an equivalent expression containing the other variable, you can create a onevariable equation that you can solve using methods shown in Chapter 2.
Vocabulary Tip
Substitution means one
value or expression is used
in place of another.
Lesson Planning and
Resources
1
See p. 372E for a list of the
resources that support this lesson.
Using Substitution
EXAMPLE
y = -4x + 8
y=x+7
Solve using substitution.
Step 1 Write an equation containing only one variable, and solve it.
PowerPoint
y = -4x + 8
Bell Ringer Practice
x + 7 = -4x + 8
5x + 7 = 8
Check Skills You’ll Need
Divide each side by 5.
Step 2 Solve for the other variable in either equation.
Lesson 3-3: Example 1
Extra Skills and Word
Problem Practice, Ch. 3
y = 0.2 + 7
Substitute 0.2 for x in y ≠ x ± 7.
y = 7.2
Simplify.
Since x = 0.2 and y = 7.2, the solution is (0.2, 7.2).
Solving Systems by Graphing
Lesson 7-1: Example 1
Extra Skills and Word
Problem Practice, Ch. 7
Check 7.2 0 -4(0.2) + 8
7.2 = 7.2 ✓
Quick Check
382
Since y ≠ x ± 7 was used in step 2, see if (0.2, 7.2)
solves y ≠ -4x ± 8.
Simplify.
1 Solve using substitution. Check your solution.
y = 2x
7x - y = 15 (3, 6)
Chapter 7 Systems of Equations and Inequalities
Special Needs
Below Level
L1
If some students have difficulty distinguishing the
colors of two different lines on a graph, redraw the
lines in two different thicknesses or have them work
with a partner.
382
Subtract 7 from each side.
x = 0.2
Equations with Variables on
Both Sides
Substitute x ± 7 for y.
Add 4x to each side.
5x = 1
For intervention, direct students to:
Start with one equation.
learning style: visual
L2
Remind students that they must check that a solution
makes all equations in the system true in order for it
to be a solution of the entire system.
learning style: verbal
2. Teach
To use the substitution method, you must have an equation that has already
been solved for one of the variables.
2
EXAMPLE
Using Substitution and the Distributive Property
Solve using the substitution method.
Guided Instruction
3y + 2x = 4
-6x + y = -7
3
-6x + y = -7
Add 6x to each side.
Step 2 Write an equation containing only one variable and solve.
3y + 2x = 4
Substitute 6x – 7 for y. Use parentheses.
18x - 21 + 2x = 4
Use the Distributive Property.
20x = 25
x = 1.25
PowerPoint
Start with the other equation.
3(6x - 7) + 2x = 4
Additional Examples
1 Solve using substitution.
y = 2x + 2
y = -3x + 4 (0.4, 2.8)
Combine like terms and add 21 to each side.
Divide each side by 20.
Step 3 Solve for the other variable in either equation.
-6 (1.25) + y = -7
-7.5 + y = -7
y = 0.5
2 Solve using substitution.
-2x + y = -1
4x + 2y = 12 (1.75, 2.5)
Substitute 1.25 for x in -6x + y ≠ -7.
Simplify.
Add 7.5 to each side.
Since x = 1.25 and y = 0.5, the solution is (1.25, 0.5).
Quick Check
2 Solve using substitution. Check your solution.
3
EXAMPLE
Real-World
6y + 8x = 28
3 = 2x - y (2.3, 1.6)
Problem Solving
Transportation Your school is planning to bring 193 people to a competition at
another school. There are eight drivers available and two types of vehicles, school
buses and minivans. The school buses seat 51 people each, and the minivans seat 8
people each. How many buses and minivans will be needed?
Let b = number of school buses.
drivers b + m = 8
Step 1 b + m = 8
51b - 8b + 64 = 193
According to the U.S.
Department of Transportation,
every year about 460,000
public school buses transport
24 million students to and
from school.
Quick Check
Substitute -b + 8 for m in the second equation.
Solve for b.
43b + 64 = 193
43b = 129
b=3
Step 3 (3) + m = 8
Ask: Why it is sometimes easier to
solve equations using substitution
rather than graphing?
Sometimes the solution is a very
large number or a decimal.
Substitute 3 for b in b + m = 8.
m=5
Three school buses and five minivans will be needed to transport 193 people.
3 Geometry A rectangle is 4 times longer than it is wide. The perimeter of the
rectangle is 30 cm. Find the dimensions of the rectangle. 3 cm by 12 cm
Lesson 7-2 Solving Systems Using Substitution
Advanced Learners
Resources
• Daily Notetaking Guide 7-2 L3
• Daily Notetaking Guide 7-2—
L1
Adapted Instruction
Closure
Subtract b from each side.
Step 2 51b + 8(-b + 8)= 193
Connection
Let m = number of minivans.
people 51b + 8m = 193
3 A youth group with 26 members is going to the beach. There
will also be five chaperones that
will each drive a van or a car. Each
van seats 7 persons, including the
driver. Each car seats 5 persons,
including the driver. How many
vans and cars will be needed?
3 vans and 2 cars
Solve the first equation for m.
m = -b + 8
Real-World
Error Prevention
Some students may equate 51b
with 51 buses. Point out that b
represents the number of buses,
and 51 represents the number of
people each bus can hold.
Step 1 Solve the second equation for y because it has a coefficient of 1.
y = 6x - 7
EXAMPLE
383
English Language Learners ELL
L4
Part of the graphs of some systems form geometric
figures. Have students write a system of three
equations whose graph forms a triangle.
learning style: visual
Some students may be confused by words such as
chaperone (as in Additional Example 3). Explain, and
discuss the word problems to verify that the students
understand what they need to do.
learning style: verbal
383
3. Practice
EXERCISES
For more exercises, see Extra Skill and Word Problem Practice.
Practiceand
andProblem
ProblemSolving
Solving
Practice
Assignment Guide
A
1 A B 1-39
C Challenge
40-41
Test Prep
Mixed Review
42-45
46-54
Practice by Example
GO for
Help
Mental Math Match each system with its solution at the right.
Example 1
1. y = x + 1 D
(page 382)
y = 2x - 1
2. y = 12 x + 4 C
2y + 2x = 2
3. 2y = x + 3 B
that they first substitute the
coordinate into the equation in
each system that is easiest to
work with, such as x = y in
Exercise 3.
Example 2
(page 383)
Example 3
(page 383)
Name
Apply Your Skills
5. y = 4x - 8
y = 2x + 10
6. C(n) = -3n - 6
C(n) = n - 4
8. y = -4x + 1212
y = 14 x + 4
9. h = 6g - 4
7. m = 5p + 8
m = -10p + 3
10. a = 25 b - 3
a = 2b - 18
h = -2g + 28
11. y = x - 2
2x + 2y = 4
12. c = 3d - 27
4d + 10c = 120
13. 3x - 6y = 30
y = -6x + 34
14. m = 4n + 11
-6n + 8m = 36
15. 7x - 8y = 112
y = -2x + 9
16. t = 0.2s + 10
4s + 5t = 35
17. Geometry The length of a rectangle is 5 cm more than twice the width. The
perimeter of the rectangle is 34 cm. Find the dimensions of the rectangle.
4 cm by 13 cm
18. Suppose you have $28.00 in your bank account and start saving $18.25 every
week. Your friend has $161.00 in his account and is withdrawing $15 every
week. When will your account balances be the same? 4 wk
Solve each system by substitution. Check your solution.
19. a - 1.2b = -3
20. 0.5x + 0.25y = 36
21. y = 0.8x + 7.2
0.2b + 0.6a = 12 (15, 15) y + 18 = 16x (9, 126)
20x + 32y = 48 (–4, 4)
L2
L1
Adapted Practice
Practice
B
L3
L4
Reteaching
D. (2, 3)
Solve each system using substitution. Check your solution. 5–16. See margin.
Exercises 1–4 Suggest to students
GPS Guided Problem Solving
C. (-2, 3)
x = 12 y + 2
x=y
Enrichment
B. (3, 3)
4. x - y = 1 A
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 6, 18, 22, 24, 32.
A. (3, 2)
Class
Practice 7-2
Date
For Exercises 22–24, define variables and write a system of equations for
each situation. Solve using substitution.
L3
Solving Systems Using Substitution
Solve each system using substitution. Write no solution or infinitely many
solutions where appropriate.
1. y = x
y = -x + 2
2. y = x + 4
y = 3x
3. y = 3x - 10
y = 2x - 5
4. x = -2y + 1
x=y-5
5. y = 5x + 5
y = 15x - 1
6. y = x - 3
y = -3x + 25
7. y = x - 7
2x + y = 8
9. x + 2y = 200
x = y + 50
10. 3x + y = 10
y = -3x + 4
11. y = 2x + 7
y = 5x + 4
12. 3x - 2y = -0
3x + 2y = -5
13. 4x + 2y = 8
y = -2x + 4
14. 6x - 3y = 6
y = 2x + 5
15. 2x + 4y = -6
2x - 3y = -7
16. 5x - 3y = -4
17. y 5 223 x 1 4
2x + 3y = -6
18. 2x + 3y = 8
3y 5 4 2 x
2
20. x + y = 0
21. 5x + 2y = 6
y 5 2 5x 1 1
2
24. y 5 2 32 x 1 1
5x + 3y = -4
19. 3x - y = 14
2x + y = 16
© Pearson Education, Inc. All rights reserved.
8. y = 3x - 6
-3x + y = -6
x=y+4
22. 2x + 5y = -6
23. 4x + 3y = -3
4x + 5y = -12
2x + 3y = -1
4x + 6y = 6
25. 5x - 6y = 19
4x + 3y = 10
26. 2x + 2y = 6.6
5x - 2y = 0.3
27. 2x - 4y = 13.8
3x - 4y = 17.7
28. 4.3x + 4y = 18
29. 3x - 4y = -5
30. y 5 31 x 1 10
x = 3y + 6
4.5x + 6y = 12
31. 2x + 5y = 62
3x - 5y = 23.3
34. 5x + 6y = -76
5x + 2y = -44
x=y+2
32. -5x + 3y = 06
-2x - 3y = 60
35. 3x - 2y = 10
y 5 3x 2 1
2
22. Agriculture A farmer grows only sunflowers and flax on his 240-acre farm.
This year he wants to plant 80 more acres of sunflowers than of flax. How many
acres of each crop does the farmer need to plant?
80 acres flax, 160 acres sunflowers
23. Renting Videos Suppose you want to join a video store. Big Video offers a
special discount card that costs $9.99 for one year. With the discount card, each
video rental costs $2.49. A discount card from Main Street Video costs $20.49
for one year. With the Main Street Video discount card, each video rental costs
$1.79. After how many video rentals is the cost the same? 15 video rentals
33. x 5 34 y 2 6
y 5 4x 1 8
3
Real-World
Connection
36. -3x + 2y = -6
37. At an ice cream parlor, ice cream cones cost $1.10 and sundaes cost
$2.35. One day, the receipts for a total of 172 cones and sundaes were
$294.20. How many cones were sold?
38. You purchase 8 gal of paint and 3 brushes for $152.50. The next day, you
purchase 6 gal of paint and 2 brushes for $113.00. How much does each
gallon of paint and each brush cost?
-2x + 2y = -6
Sunflower seeds are sold
as snacks and as bird food,
and they are a source of
cooking oil.
384
Chapter 7 Systems of Equations and Inequalities
pages 384–386
Exercises
5– 16. Coordinates given in
alphabetical order.
5. (9, 28)
6. –21 , –4 21
(
384
24. Buying a Car Suppose you are thinking about buying one of two cars. Car A
GPS will cost $17,655. You can expect to pay an average of $1230 per year for fuel,
maintenance, and repairs. Car B will cost about $15,900. Fuel, maintenance, and
repairs for it will average about $1425 per year. After how many years are the
total costs for the cars the same? 9 yr
)
7. (6 31 , –31
8. 2, 4 12
9. (4, 20)
10. 43 , 9 38
11. (2, 0)
(
)
(
)
)
12.
13.
14.
15.
16.
(7 177 , 11 178 )
(6, –2)
(3, –2)
(8, –7)
(–3, 9.4)
GO
nline
Homework Help
Visit: PHSchool.com
Web Code: ate-0702
25. Multiple Choice You have 28 coins that are all nickels n and dimes d. The value
of the coins is $2.05. Which system of equations can be used to find the number
of nickels and the number of dimes? D
n + d = 28
n + d = 205
10n + 5d = 2.05
n + d = 28
10n + 5d = 205
n + d = 28
n + d = 28
5n + 10d = 205
Estimation Graph each system to estimate the solution. Then use substitution to
find the exact solution of the system. 26–31. See back of book.
32. Answers may vary.
Sample: y ≠ x and
y ≠ –3x ± 2; (12 , 12 )
26. y = 2x
y = -6x + 4
27. y = 12 x + 4
y = - 4x - 5
28. x + y = 0
5x + 2y = -3
29. y = 2x + 3
y = 0.5x - 2
30. y = -x + 4
y = 2x + 6
31. y = 0.7x + 3
y = -1.5x - 7
32. Open-Ended Write a system of linear equations with exactly one solution. Use
substitution to solve your system. See left.
33. a. Solve each system below using substitution. (x, y) such that y ≠ 0.5x ± 4;
no solution
y = 0.5x + 4
6x - 2y = 10
-x + 2y = 8
Error Prevention!
Exercise 17 Some students may
use p = 34 for the second
equation. Explain to students that
the expression for finding
perimeter, 2/ + 2w, must be used
so that the equations have the
same variables.
Alternative Method
Exercise 18 Students may want to
solve this problem by keeping
two running balances like those
used in bank account registers.
Have them make side-by-side
vertical lists to find when the
numbers are the same.
Week
$28 ± 18.25
$161 – 15
1
2
46.25
64.50
etc.
101.00
146.00
131.00
etc.
101.00
y = 3x + 1
b. Solve each system by graphing. See back of book.
c. Critical Thinking Make a general statement about the solutions you
get when solving by graphing and the results you get when solving
by substitution. See margin.
46.
12
Solve each system using substitution.
C
Challenge
6
34. y = 2x
6x - y = 8 (2, 4)
35. y = 3x + 1
x = 3y + 1 (–21 , –12 )
36. x - 3y = 14
x - 2 = 0 (2, –4)
37. 2x + 2y = 5
38. 4x + y = -2
39. 3x + 5y = 2
y = 14 x
(2, )
1
2
-2x - 3y = 1 (
–12 ,
0)
6 O
40. There are 1170 students in a school. The ratio of girls to boys is 23 i 22.
The system below describes relationships between the number of girls
and the number of boys.
g
23
g + b = 1170
b = 22
47.
12
x
18
y
4
2
x
O
41. Sprinting The graph at the left represents the start of a 100-meter race
between Joetta and Gail. The red line and blue line represent Joetta’s and
Gail’s time and distance. Joetta averages 8.8 m/s. Gail averages 9 m/s but
started 0.2 s after Joetta. At time 0.2, Gail’s distance is 0 m. You can use pointslope form to write an equation that relates Gail’s time t to her distance d.
Distance
(meters)
6
x + 4y = -4 (4, –2)
a. Solve the proportion for g. g ≠ 23
22 b
b. Solve the system. (b, g) ≠ (572, 598)
c. How many more girls are there than boys? 26
48.
d = 9t - 1.8
2
y
2
O
y - y1 = m(x - x1 )
d - 0 = 9(t - 0.2)
Time
(seconds)
y
49.
Since Joetta started at t = 0, the equation d = 8.8t relates her time and distance.
a. Solve the system using substitution. (t, d) ≠ (9, 79.2)
b. Will Gail overtake Joetta before the finish line? yes
2
2
4
x
y
x
O
2
2
lesson quiz, PHSchool.com, Web Code: ata-0702
Lesson 7-2 Solving Systems Using Substitution
33. c. When there is exactly one solution, the graph will show
two lines that intersect in a single point, and the
substitution method will give the coordinates of the
point. When there are infinitely many solutions, the
graphs coincide, and substitution leads to an equation
that is always true. When there are no solutions, the
graphs are parallel, and substitution results in an
equation that is never true.
385
45.[2] 7(–7) – 4(–2) 29
–49 ± 8 29
–49 29
No, (–2, –7) must
satisfy both equations
to be a solution of the
system.
[1] no explanation given
50.
f(x)
2
x
O
2
385
4. Assess & Reteach
Standardized
Test PrepTest Prep
Gridded Response
PowerPoint
Lesson Quiz
Solve each system using
substitution.
42. Find the value of the y-coordinate
of the solution to the given system.
5x + 5y = 179
x = 5y - 143 29.8
43. Find the value of the y-coordinate
of the solution to the given system.
y = 9x + 3480
y = 81x - 7104 4803
44. Tina has $220 in her account. Cliff has $100 in his account. Starting in July,
Tina adds $25 to her account on the first of each month, while Cliff adds
$35 to his. How many dollars will they have in their accounts when the
amounts are the same? 520
1. 5x + 4y = 5
y = 5x (0.2, 1)
2. 3x + y = 4
2x - y = 6 (2, –2)
45. Is (-2, -7) the solution of the following system? Justify your answer.
7y - 4x = 29 See margin.
x=y-5
Short Response
3. 6m - 2n = 7
3m + n = 4 (1.25, 0.25)
Alternative Assessment
Group students in pairs. Give each
student a system of equations.
One partner solves the system
using a graphing calculator. The
other student uses substitution to
solve the system. They compare
answers to check. Have partners
switch roles.
Mixed Review
GO for
Help
Solve each system by graphing. 46–48. See margin for graphs.
Lesson 7-1
46. y = x - 2
y = 12 x + 4 (12, 10)
Test Prep
y = 14 x + 1 (4, 2)
49. y = 3x - 2
50. f(x) = x + 1
51. f(x) = -2x
52. y = Δx« + 5
53. y = -2Δx«
54. y = Δx + 3« - 1
Checkpoint Quiz 1
Checkpoint Quiz 1
Lessons
Lessons 7-1
7-1 through
through 7-2
7-2
Solve each system by graphing. 1–3. See margin for graphs.
y = 23 x (3, 2)
y = -2x + 1 (1, –1)
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 425
• Test-Taking Strategies, p. 420
• Test-Taking Strategies with
Transparencies
3. y = 14 x - 1
2. y = 43 x - 2
1. y = 3x - 4
Resources
y = -2x - 10
(–4, –2)
Solve each system using substitution.
4. y = 3x - 14
y = x - 10 (2, –8)
5. y = 2x + 5
y = 6x + 1 (1, 7)
7. 3x + 4y = 12
y = -2x + 10 (
8. 4x + 9y = 24
28
5 ,
–65
)
6. x = y + 7
y = 8 + 2x
(–15, –22)
y = 2 13 x + 2 (6, 0)
In Exercises 9 and 10, write and solve a system of equations for each situation.
Use this Checkpoint Quiz to check
students’ understanding of the
skills and concepts of Lessons 7-1
and 7-2.
9. A rectangle is 3 times longer than it is wide. The perimeter is 44 cm. Find the
dimensions of the rectangle. 2L ± 2W ≠ 44, 3W ≠ L; 5.5 cm by 16.5 cm
10. A farmer grows only pumpkins and corn on her 420-acre farm. This year she
wants to plant 250 more acres of corn than of pumpkins. How many acres of
each crop does the farmer need to plant? p ± c ≠ 420, c ≠ p ± 250; 85 acres
pumpkins, 335 acres corn
Resources
Grab & Go
• Checkpoint Quiz 1
386
Chapter 7 Systems of Equations and Inequalities
Checkpoint Quiz
1.
y
x
O
2
386
y = -x + 3 (2, 1)
Graph each function. 49–50. See margin. 51–54. See back of book.
Lesson 5-3
A sheet of blank grids is available
with the Test-Taking Strategies
booklet. Give this sheet to
students for practice with filling
in the grids.
48. y = 34 x - 1
47. y = -2x + 5
2
2.
3.
y
2
y
x
4
x
O
2
4
2 O
2