The document discusses various methods for solving systems of equations, including substitution, elimination, graphic, and matrix methods. It provides an example of using the substitution method to solve the system of equations 5x + y = 2 and 3x - 2y = 22. The solution obtained is x = 2 and y = -8.

1. Chapter 9
Systems of Equations & Inequalities
Hebrews 12:2 "Looking to Jesus, the founder and perfecter of
our faith, who for the joy that was set before him endured
the cross, despising the shame, and is seated at the right hand
of the throne of God."

2. 9.1 Systems of Equations
A system of equations is a set of equations
that involve the same variables.

3. 9.1 Systems of Equations
A system of equations is a set of equations
that involve the same variables.
Number of Variables = Number of Equations Needed

4. Methods of Solving a Systems Problem:

5. Methods of Solving a Systems Problem:
1. Substitution Method

6. Methods of Solving a Systems Problem:
1. Substitution Method
2. Elimination Method

7. Methods of Solving a Systems Problem:
1. Substitution Method
2. Elimination Method
3. Graphic Method

8. Methods of Solving a Systems Problem:
1. Substitution Method
2. Elimination Method
3. Graphic Method
4. Matrix Method

9. Methods of Solving a Systems Problem:
1. Substitution Method
2. Elimination Method
3. Graphic Method
4. Matrix Method
Last year you learned A B
−1

10. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution

11. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2
⎨
⎩ 3x − 2y = 22

12. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.

13. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x

14. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x
2. 3x − 2 ( 2 − 5x ) = 22

15. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x
2. 3x − 2 ( 2 − 5x ) = 22
3x − 4 + 10x = 22

16. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x
2. 3x − 2 ( 2 − 5x ) = 22
3x − 4 + 10x = 22
13x = 26

17. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x
2. 3x − 2 ( 2 − 5x ) = 22
3x − 4 + 10x = 22
13x = 26
x=2

18. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x 1. y = 2 − 5 ( 2 )
2. 3x − 2 ( 2 − 5x ) = 22
3x − 4 + 10x = 22
13x = 26
x=2

19. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x 1. y = 2 − 5 ( 2 )
2. 3x − 2 ( 2 − 5x ) = 22 y = 2 − 10
3x − 4 + 10x = 22
13x = 26
x=2

20. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x 1. y = 2 − 5 ( 2 )
2. 3x − 2 ( 2 − 5x ) = 22 y = 2 − 10
3x − 4 + 10x = 22
y = −8
13x = 26
x=2

21. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Substitution
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x 1. y = 2 − 5 ( 2 )
2. 3x − 2 ( 2 − 5x ) = 22 y = 2 − 10
3x − 4 + 10x = 22
y = −8
13x = 26
x=2 x = 2, y = −8

22. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination

23. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.

24. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
⎧10x + 2y = 4
⎨
⎩ 3x − 2y = 22

25. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
⎧10x + 2y = 4
⎨
⎩ 3x − 2y = 22

26. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
⎧10x + 2y = 4
⎨
⎩ 3x − 2y = 22
13x = 26

27. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
⎧10x + 2y = 4
⎨
⎩ 3x − 2y = 22
13x = 26
x=2

28. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨ 1. 5 ( 2 ) + y = 2
⎩ 3x − 2y = 22 2.
⎧10x + 2y = 4
⎨
⎩ 3x − 2y = 22
13x = 26
x=2

29. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨ 1. 5 ( 2 ) + y = 2
⎩ 3x − 2y = 22 2.
y = 2 − 10
⎧10x + 2y = 4
⎨
⎩ 3x − 2y = 22
13x = 26
x=2

30. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨ 1. 5 ( 2 ) + y = 2
⎩ 3x − 2y = 22 2.
y = 2 − 10
⎧10x + 2y = 4
⎨ y = −8
⎩ 3x − 2y = 22
13x = 26
x=2

31. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Elimination
⎧5x + y = 2 1.
⎨ 1. 5 ( 2 ) + y = 2
⎩ 3x − 2y = 22 2.
y = 2 − 10
⎧10x + 2y = 4
⎨ y = −8
⎩ 3x − 2y = 22
13x = 26 x = 2, y = −8
x=2

32. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic

33. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.

34. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x

35. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x
2. − 2y = 22 − 3x

36. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1.
⎨
⎩ 3x − 2y = 22 2.
1. y = 2 − 5x
2. − 2y = 22 − 3x
22 − 3x
y=
−2

37. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1. On your calculator:
⎨ y1 = 2 − 5x
⎩ 3x − 2y = 22 2.
y2 = ( 22 − 3x ) / ( −2 )
1. y = 2 − 5x
2. − 2y = 22 − 3x
22 − 3x
y=
−2

38. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1. On your calculator:
⎨ y1 = 2 − 5x
⎩ 3x − 2y = 22 2.
y2 = ( 22 − 3x ) / ( −2 )
1. y = 2 − 5x
Graph and find all
2. − 2y = 22 − 3x
points of intersection
22 − 3x
y=
−2

39. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Graphic
⎧5x + y = 2 1. On your calculator:
⎨ y1 = 2 − 5x
⎩ 3x − 2y = 22 2.
y2 = ( 22 − 3x ) / ( −2 )
1. y = 2 − 5x
Graph and find all
2. − 2y = 22 − 3x
points of intersection
22 − 3x
y=
−2 x = 2, y = −8

40. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Matrices

41. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Matrices
On your calculator
enter the matrices:
⎡ 5 1 ⎤ ⎡ 2 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 −2 ⎦ ⎣ 22 ⎦

42. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Matrices
On your calculator
enter the matrices:
⎡ 5 1 ⎤ ⎡ 2 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 −2 ⎦ ⎣ 22 ⎦
then calculate A B
−1

43. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Matrices
On your calculator
enter the matrices:
⎡ 5 1 ⎤ ⎡ 2 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 −2 ⎦ ⎣ 22 ⎦
then calculate A B−1
⎡ 2 ⎤
⎢ ⎥
⎣ −8 ⎦

44. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Matrices
On your calculator
enter the matrices:
⎡ 5 1 ⎤ ⎡ 2 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 −2 ⎦ ⎣ 22 ⎦
then calculate A B−1
⎡ 2 ⎤
⎢ ⎥ x = 2, y = −8
⎣ −8 ⎦

45. ⎧5x + y = 2
Solve ⎨ using each method.
⎩ 3x − 2y = 22
Matrices
On your calculator Note: the matrix
enter the matrices: method works only
for systems of linear
⎡ 5 1 ⎤ ⎡ 2 ⎤
A = ⎢ ⎥ B = ⎢ ⎥ equations!!!
⎣ 3 −2 ⎦ ⎣ 22 ⎦
then calculate A B−1
⎡ 2 ⎤
⎢ ⎥ x = 2, y = −8
⎣ −8 ⎦

46. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25

47. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
We have a circle and a line ... how could they intersect?

48. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
We have a circle and a line ... how could they intersect?
2 points

49. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
We have a circle and a line ... how could they intersect?
2 points 1 point

50. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
We have a circle and a line ... how could they intersect?
2 points 1 point 0 points

51. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25

52. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25

53. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25
2 2
1. x + ( 3x + 25 ) = 625

54. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25
2 2
1. x + ( 3x + 25 ) = 625
2 2
x + 9x + 150x + 625 = 625

55. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25
2 2
1. x + ( 3x + 25 ) = 625
2 2
x + 9x + 150x + 625 = 625
10x 2 + 150x = 0

56. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25
2 2
1. x + ( 3x + 25 ) = 625
2 2
x + 9x + 150x + 625 = 625
10x 2 + 150x = 0
10x ( x + 15 ) = 0

57. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25
2 2
1. x + ( 3x + 25 ) = 625
2 2
x + 9x + 150x + 625 = 625
10x 2 + 150x = 0
10x ( x + 15 ) = 0
x1 = 0 x2 = −15

58. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625
2 2
x + 9x + 150x + 625 = 625
10x 2 + 150x = 0
10x ( x + 15 ) = 0
x1 = 0 x2 = −15

59. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625 y1 = 25
2 2
x + 9x + 150x + 625 = 625
10x 2 + 150x = 0
10x ( x + 15 ) = 0
x1 = 0 x2 = −15

60. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625 y1 = 25
2 2
x + 9x + 150x + 625 = 625
2. y2 = 3( −15 ) + 25
10x 2 + 150x = 0
10x ( x + 15 ) = 0
x1 = 0 x2 = −15

61. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625 y1 = 25
2 2
x + 9x + 150x + 625 = 625
2. y2 = 3( −15 ) + 25
10x 2 + 150x = 0
y2 = −45 + 25
10x ( x + 15 ) = 0
x1 = 0 x2 = −15

62. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625 y1 = 25
2 2
x + 9x + 150x + 625 = 625
2. y2 = 3( −15 ) + 25
10x 2 + 150x = 0
y2 = −45 + 25
10x ( x + 15 ) = 0
y2 = −20
x1 = 0 x2 = −15

63. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625 y1 = 25
2 2
x + 9x + 150x + 625 = 625
2. y2 = 3( −15 ) + 25
10x 2 + 150x = 0
y2 = −45 + 25
10x ( x + 15 ) = 0
y2 = −20
x1 = 0 x2 = −15
( 0, 25 ), ( −15, − 20 )

64. 2 2
⎧ x + y = 625
Solve ⎨ using Substitution.
⎩ 3x − y = −25
2. y = 3x + 25 2. y1 = 3( 0 ) + 25
2 2
1. x + ( 3x + 25 ) = 625 y1 = 25
2 2
x + 9x + 150x + 625 = 625
2. y2 = 3( −15 ) + 25
10x 2 + 150x = 0
y2 = −45 + 25
10x ( x + 15 ) = 0
y2 = −20
x1 = 0 x2 = −15
( 0, 25 ), ( −15, − 20 )
We usually show these solutions as ordered pairs

65. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩

66. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas

67. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩

68. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩
11y = −66

69. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩
11y = −66
y = −6

70. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩
11y = −66
y = −6
2
2. 4x + 3( −6 ) = 18

71. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩
11y = −66
y = −6
2
2. 4x + 3( −6 ) = 18
2
4x − 18 = 18

72. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩
11y = −66
y = −6
2
2. 4x + 3( −6 ) = 18
2
4x − 18 = 18
2
4x = 36

73. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩ 2
x =9
11y = −66
y = −6
2
2. 4x + 3( −6 ) = 18
2
4x − 18 = 18
2
4x = 36

74. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩ 2
x =9
11y = −66 x=±3
y = −6
2
2. 4x + 3( −6 ) = 18
2
4x − 18 = 18
2
4x = 36

75. 2
⎧ 3x + 5y = −3
⎪
Solve ⎨ using Elimination.
2
⎪ 4x + 3y = 18
⎩ 2 parabolas
⎧12x 2 + 20y = −12
⎪
⎨ 2
⎪−12x − 9y = −54
⎩ 2
x =9
11y = −66 x=±3
y = −6
2
2. 4x + 3( −6 ) = 18
( 3, − 6 ), ( −3, − 6 )
2
4x − 18 = 18
2
4x = 36

76. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩

77. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle

78. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x

79. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x
2 2
2. y = 2 − x

80. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x
2 2
2. y = 2 − x
2
y= ± 2− x

81. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x
2 2
2. y = 2 − x
2
y= ± 2− x
2
y2 = 2 − x

82. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x
2 2
2. y = 2 − x
2
y= ± 2− x
2
y2 = 2 − x
2
y3 = − 2 − x

83. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x
2 2
2. y = 2 − x
2
y= ± 2− x
2
y2 = 2 − x
(1.2, .7 ), ( −1.2, − .7 )
2
y3 = − 2 − x

84. 3
⎧ y = x − x
⎪
Solve ⎨ graphically.
2 2
⎪ x + y = 2
⎩ a cubic and a circle
3
1. y1 = x − x
2 2
2. y = 2 − x
2
y= ± 2− x
2
y2 = 2 − x
(1.2, .7 ), ( −1.2, − .7 )
2
y3 = − 2 − x
it is not correct to state this as the solution: ( ± 1.2, ± .7 )
as this has four combinations ...

85. HW #1
Your time is limited, so don’t waste it living someone
else’s life.
Steve Jobs