Solve Systems by Adding, Subtracting, and Multiplying

1. Section 8-4
Solve Systems by Adding, Subtracting, and
Multiplying (Linear Combinations)

2. Essential Questions
How do you solve systems of equations
by adding and subtracting?
How do you solve systems of equations
by adding and multiplying?
Where you’ll see this:
Landscaping, construction, sports

3. Why another method?

4. Why another method?
Graphing: Problems with accuracy

5. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable

6. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Combinations: Could speed up
process

7. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Combinations: Could speed up
process
Look at the system to determine best method

8. Solve by Combinations

9. Solve by Combinations
1. Choose a variable to eliminate (your choice).

10. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.

11. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
3. Solve for the remaining variable.

12. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
3. Solve for the remaining variable.
4. Plug back into an original equation to find
the other variable.

13. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
3. Solve for the remaining variable.
4. Plug back into an original equation to find
the other variable.
5. Check and rewrite the answer.

14. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18


15. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2

16. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
2x + 7 y = 18

17. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
2x + 7 y = 18

18. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
2x + 7 y = 18
10 y = 20

19. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
2x + 7 y = 18
10 y = 20
10 10

20. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
2x + 7 y = 18
10 y = 20
10 10
y=2

21. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
10 y = 20
10 10
y=2

22. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20
10 10
y=2

23. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10
y=2

24. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2

25. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2

26. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18

−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2

27. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18
 Check:
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2

28. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18
 Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2

29. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18
 Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2

30. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18
 Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6 2(2) + 7(2) = 18
10 10 −2x = −4
y=2 −2 −2
x=2

31. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18
 Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6 2(2) + 7(2) = 18
10 10 −2x = −4 4 + 14 = 18
y=2 −2 −2
x=2

32. Example 1
Solve by combining the equations
−2x + 3 y = 2

a. 
 2x + 7 y = 18
 Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6 2(2) + 7(2) = 18
10 10 −2x = −4 4 + 14 = 18
y=2 −2 −2
x=2 (2, 2)

33. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
x + 4 y = 5


34. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(

35. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14

36. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5

37. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5

38. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5
9x = 9

39. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5
9x = 9
9 9

40. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5
9x = 9
9 9
x=1

41. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9
9 9
x=1

42. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9
x=1

43. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
x=1

44. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1

45. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
y =1

46. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
y =1

47. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
y =1

48. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4
x=1
y =1

49. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4 1 + 4(1) = 5
x=1
y =1

50. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4 1 + 4(1) = 5
x=1 1+ 4 = 5
y =1

51. Example 1
Solve by combining the equations
10x + 4 y = 14

b. 
 x + 4 y = 5)(−1)
( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4 1 + 4(1) = 5
x=1 1+ 4 = 5
y =1
(1,1)

52. Example 2
Solve by combining the equations
7x + 2 y = 5


 2x + 3 y = 16


53. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16


54. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(

55. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15

56. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15
−4x − 6 y = −32

57. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15
−4x − 6 y = −32

58. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15
−4x − 6 y = −32
17x = −17

59. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15
−4x − 6 y = −32
17x = −17
17 17

60. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15
−4x − 6 y = −32
17x = −17
17 17
x = −1

61. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32
17x = −17
17 17
x = −1

62. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17
17 17
x = −1

63. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17
x = −1

64. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1

65. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3

66. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6

67. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6

68. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6

69. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
−7 + 12 = 5
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6

70. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
−7 + 12 = 5
17x = −17 +2 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3
y=6

71. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
−7 + 12 = 5
17x = −17 +2 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3 −2 + 18 = 16
y=6

72. Example 2
Solve by combining the equations
7x + 2 y = 5)(3)
(

 2x + 3 y = 16)(−2)
(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
−7 + 12 = 5
17x = −17 +2 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3 −2 + 18 = 16
y=6
(−1, 6)

73. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3


74. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(

75. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7

76. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7
−4x − 2 y = 3

77. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7
−4x − 2 y = 3

78. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7
−4x − 2 y = 3
x = 10

79. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3
x = 10

80. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
x = 10

81. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10

82. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43

83. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
2 2

84. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
2 2
−43
y=
2

85. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
Check:
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
2 2
−43
y=
2

86. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 −50
x = 10
2 y = −43
2 2
−43
y=
2

87. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43
2 2
−43
y=
2

88. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43 4(10) + 2(− ) = −3
43
2
2 2
−43
y=
2

89. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43 4(10) + 2(− ) = −3
43
2
2 2 40 − 43 = −3
−43
y=
2

90. Example 3
Solve by combining the equations
5x + 2 y = 7


4x + 2 y = −3)(−1)
(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43 4(10) + 2(− ) = −3
43
2
2 2 40 − 43 = −3
−43
y=
2
(10, − ) 43
2

91. Homework

92. Homework
p. 350 #1-27 odd
“A verbal contract isn’t worth the paper it’s
written on.” - Samuel Goldwyn