Solve Systems by Adding, Subtracting, and Multiplying (Linear Combinations)

1. Section 8-4
Solve Systems by Adding, Subtracting, and
Multiplying (Linear Combinations)
Sun, Apr 10

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
Sun, Apr 10

3. Why another method?
Sun, Apr 10

4. Why another method?
Graphing: Problems with accuracy
Sun, Apr 10

5. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Sun, Apr 10

6. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Combinations: Could speed up
process
Sun, Apr 10

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
Sun, Apr 10

8. Solve by Combinations
Sun, Apr 10

9. Solve by Combinations
1. Choose a variable to eliminate (your choice).
Sun, Apr 10

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.
Sun, Apr 10

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.
Sun, Apr 10

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.
Sun, Apr 10

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.
Sun, Apr 10

14. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
Sun, Apr 10

15. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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)
Sun, Apr 10

33. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪x + 4 y = 5
⎩
Sun, Apr 10

34. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
Sun, Apr 10

35. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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)
Sun, Apr 10

52. Example 2
Solve by combining the equations
⎧7x + 2 y = 5
⎪
⎨
⎪ 2x + 3 y = 16
⎩
Sun, Apr 10

53. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16
⎩
Sun, Apr 10

54. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Sun, Apr 10

55. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18
x = −1 3 3
y=6
Sun, Apr 10

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
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3
y=6
Sun, Apr 10

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
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3 −2 + 18 = 16
y=6
Sun, Apr 10

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
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3 −2 + 18 = 16
y=6
(−1, 6)
Sun, Apr 10

73. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3
⎩
Sun, Apr 10

74. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Sun, Apr 10

75. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 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
Sun, Apr 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
Sun, Apr 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
Sun, Apr 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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

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
Sun, Apr 10

91. Problem Set
Sun, Apr 10

92. Problem Set
p. 350 #1-27 odd
“A verbal contract isn’t worth the paper it’s
written on.” - Samuel Goldwyn
Sun, Apr 10