Download 6-4 Solve Equivalent Systems of Linear Equations with the Same

6-4 Solve Equivalent Systems of Linear Equations
with the Same Solution
Name
Date
Solve: ⎧ 4x 5y 7
⎨
⎩ 6x 7y 33
Add the equations.
7(4x 5y 7)
5(6x 7y 33)
28x 35y 49
30x 35y 165
58x
116
58x 116
58
58
Use the Multiplication
Property of Equality.
Use the Division Property
of Equality.
Substitute 2 for x in one of the original
equations. Solve for y.
4(2) 5y 7
Solve for y by substituting
2 for x.
8 5y 7
8 8 5y 7 8
5y 15
5y 5 15 5
x2
Use the Division
Property of Equality.
y 3
Check: Substitute 2 for x and 3 for y in both of the original equations to check.
4x 5y 7
?
4(2) 5(3) 7
?
8 15 7
7 7 True
6x 7y 33
?
6(2) 7(3) 33
?
12 21 33
33 33 True
So the solution of the system of equations is (2, 3).
Solve each system of equations. Check your answer on a separate sheet of paper.
Copyright © by William H. Sadlier, Inc. All rights reserved.
1. ⎧ 3x 2y 22
⎨
⎩ 2x 4y 28
2(3x 2y 22)
2x 4y 28
6x 4y 44
2x 4y 28
4x
–16
x4
3(4) 2y 22
2y 10
y 5; (4, 5)
5. ⎧ 8c 2d 1
⎨
⎩ 12c 4d 1
16c 4d 2
12c 4d 1
4c
1
1
c
4
1
8
2d 1
4
1
2d 1; d 2
1 1
,
4 2
()
(
)
2. ⎧ 5x 3y 42
⎨
⎩ 2x 6y 60
2(5x 3y 42)
2x 6y 60
10x 6y 84
2x 6y 60
8x
–24
x3
5(3) 3y 42
3y 27
y 9; (3, 9)
6. ⎧ 9a 6b 1
⎨
⎩ 6a 9b 1
3. ⎧ 3x 5y 36
⎨
⎩ 5x 2y 2
4. ⎧ 6x 4y 30
⎨
⎩ 2x 5y 29
2(3x 5y 36)
5(5x 2y 2)
6x 10y 72
25x 10y 10
31x
62
x 2
3(–2) – 5y 36
5y 30
y 6; (2, 6)
6x 4y 30
3(2x 5y 29)
6x 4y 30
6x 15y 87
19y 57
y 3
2x 5(–3) 29
2x 14; x 7
7. ⎧ 14 8s 10r
⎨
⎩ 5r 5 12s
8. ⎧ 10 12q 6p
⎨
⎩ 24p 37 18q
(7, 3)
14 8s 10r
24p 48q 40
18a 12b 2
10 24s 10r
24p 18q 37
18a 27b 3
24 32s 0
66q 77
15b 5
3
7
1
24
32s;
s
q
b
4
6
3
3
7
1
5r 5 12
10 12 6p
6a 9
1
4
6
3
4
1
5r –4; r 4 6p
6a 2; a 5
3
2 7
2
1 1
4 3
p ; ,
,
,
3 6
3
3 3
5 4
()
()
(
)
(
Lesson 6-4, pages 158–159.
)
( )
(
Chapter 6
)
149
For More Practice Go To:
Solve each system of equations. Check your answer on a separate sheet of paper.
1
2
10. ⎧ 13.8x 9.2y 4
11. ⎧⎪ 2 x 3 y 8
9. ⎧ 3x 5y 12
⎨
⎨
⎨ 3
⎩ 12.6x 21y 1
⎩ 6x 4y 8
⎪ x 1y 22
5
5
⎩ 20
21(3x 5y 12)
5(12.6x 21y 1)
63x 105y 252
(63x 105y 5)
0 247 False
no solution
4(13.8x 9.2y 4)
9.2(6x 4y 8)
55.2x 36.8y 16
(55.2x 36.8y 73.6)
0 89.6 False
no solution
6x 8y 96
2(3x 4y 48)
6x 8y 96
(6x 8y 96)
0 0 True
all real numbers
Solve. Show your work.
Let a apple and o orange.
⎧ 3a 2o 0.65
Solve: ⎨
⎩ 2a 5o 0.8
2(3a 2o 0.65) 6a 4o 1.3
3(2a 5o 0.8) (6a 15o 2.4)
11o 1.1; o 0.1
Solve for a: 3a 2(0.1) 0.65; 3a 0.2 0.65
3a 0.45; a 0.15. An apple costs $0.15.
An orange costs $0.10.
14. Geometry Two angles are complementary.
One angle is 2° less than 3 times the other
angle. Find the measure of each angle.
(Hint: The sum of the measures of
complementary angles is 90°.)
Let x, y complementary angles.
⎧ x y 90
Solve: ⎨
⎩ y 3x 2
3(x y 90)
3x 3y 270
(3x y 2)
y 3x 2
4y 268; y 67
Solve for x: x 67 90; x 23
The measures of the angles are 23° and 67°.
13. On Monday, Ming works 4 h 30 min, and
Dahlia works 3.5 h. Together, they earn $78.15.
On Tuesday, Ming works 5 h 12 min, and
Dahlia works 4.8 h. Together, they earn $98.92.
What is each person’s hourly rate?
Let m Ming’s hourly rate, d Dahlia’s hourly rate.
⎧ 4.5m 3.5d 78.15
Solve: ⎨
⎩ 5.2m 4.8d 98.92
5.2(4.5m 3.5d 78.15)
4.5(5.2m 4.8d 98.92)
23.4m 18.2d 406.38
23.4m 21.6d 445.14
3.4d 38.76; d 11.4
Solve for m: 4.5m 3.5(11.4) 78.15
4.5m 39.9 78.15; 4.5m 38.25; m 8.5
Ming makes $8.50/h. Dahlia makes $11.40/h.
15. Geometry Two angles are supplementary.
One angle is 15° less than 2 times the other
angle. Find the measure of each angle.
(Hint: The sum of the measures of
supplementary angles is 180°.)
Let a, b supplementary angles.
⎧ a b 180
Solve: ⎨
⎩ a 2b 15
2a 2b 360
2(a b 180)
a 2b 15
a 2b 15
3a
345; a 115
Solve for b: 115 b 180; b 65
The measures of the angles are 115° and 65°.
16. By which variables should you multiply each equation in the system
of equations to eliminate x?
Explain your answer.
⎧ ax by c
⎨
⎩ dx ey f
Answers may vary. Check students’ responses. Possible response:
Multiply the first equation by d and the second equation by a.
⎧ adx bdy cd
This will give the system ⎨ adx (ae)y af . Then add the equations to eliminate x.
⎩
150
Chapter 6
Copyright © by William H. Sadlier, Inc. All rights reserved.
12. Three apples and 2 oranges cost $0.65. Two
apples and 5 oranges cost $0.80. What is the
cost of 1 apple and 1 orange?