Solving Systems by Substitution

1. 1. b 2. a 3. a. (6, 20); b. Plug into both equations
4. (2, 12) 5. (-8/3, -19,3) 6. b 7. It is a real world situation
8. a. Substitute to see answers are close; b. Substitute again
9. Advantage: Can see total number of solutions
Disadvantage: Tough to find exact answer
10. a. 0; b. n/a; c. n/a 12. a. 1; b. (2, 4); c. Plug in
14. a. Graph; b. 3 solutions; c. (-3.8, .2), (-1, 3), (.8, 4.8)
16. Graph the two equations; they do not intercept
18. 20. Whole numbers ≥ 21
-1 0 1 2 3 4 5
2
T − 2π r 24. x = 11 or x = -11
22. h =
2π r

2. Section 5-3
Solving Systems by Substitution

3. Warm-up
1. Solve 8x + 8(5-2x) = -40
2. Evaluate 3x - 2 when x = 4y + 1

4. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
2. Evaluate 3x - 2 when x = 4y + 1

5. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
2. Evaluate 3x - 2 when x = 4y + 1

6. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
-8x + 40 = -40
2. Evaluate 3x - 2 when x = 4y + 1

7. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
-8x + 40 = -40
-8x = -80
2. Evaluate 3x - 2 when x = 4y + 1

8. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
-8x + 40 = -40
-8x = -80
x = 10
2. Evaluate 3x - 2 when x = 4y + 1

9. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
-8x + 40 = -40
-8x = -80
x = 10
2. Evaluate 3x - 2 when x = 4y + 1
3(4y + 1) - 2

10. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
-8x + 40 = -40
-8x = -80
x = 10
2. Evaluate 3x - 2 when x = 4y + 1
3(4y + 1) - 2
12y + 3 - 2

11. Warm-up
1. Solve 8x + 8(5-2x) = -40
8x + 8(5-2x) = -40
8x + 40 -16x = -40
-8x + 40 = -40
-8x = -80
x = 10
2. Evaluate 3x - 2 when x = 4y + 1
3(4y + 1) - 2
12y + 3 - 2
12y + 1

14. 1. Tables

15. 1. Tables Not very efficient

16. 1. Tables Not very efficient
2. Graphing by hand

17. 1. Tables Not very efficient
2. Graphing by hand Not very accurate

18. 1. Tables Not very efficient
2. Graphing by hand Not very accurate
3. Graphing Calculator

19. 1. Tables Not very efficient
2. Graphing by hand Not very accurate
3. Graphing Calculator Cheap way out

20. Example 1
Solve.
x + y = 6

y = x + 2


21. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6

22. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6
2x + 2 = 6

23. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6
2x + 2 = 6
2x = 4

24. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6
2x + 2 = 6
2x = 4
x=2

25. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2
2x + 2 = 6
2x = 4
x=2

26. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2
2x + 2 = 6 y=2+2
2x = 4
x=2

27. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2
2x + 2 = 6 y=2+2
2x = 4 y=4
x=2

28. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2 x+y=6
2x + 2 = 6 y=2+2
2x = 4 y=4
x=2

29. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2 x+y=6
2x + 2 = 6 y=2+2 2+4=6
2x = 4 y=4
x=2

30. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2 x+y=6
2x + 2 = 6 y=2+2 2+4=6
2x = 4 y=4
x=2 (2, 4)

31. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2 x+y=6
2x + 2 = 6 y=2+2 2+4=6
2x = 4 y=4
x=2 (2, 4)
Always check your answer.

32. Example 1
Solve.
x + y = 6

y = x + 2

x + (x + 2) = 6 y=x+2 x+y=6
2x + 2 = 6 y=2+2 2+4=6
2x = 4 y=4
x=2 (2, 4)
Always check your answer.
You’ll know you’re right.

33. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.

34. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets

35. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets






36. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750





37. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750

 S = 2A



38. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750

 S = 2A

 C = 1/2 A

39. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750 A + 2A + 1/2 A = 1750

 S = 2A

 C = 1/2 A

40. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750 A + 2A + 1/2 A = 1750

 S = 2A 7/2 A = 1750

 C = 1/2 A

41. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750 A + 2A + 1/2 A = 1750

 S = 2A 7/2 A = 1750

 C = 1/2 A A = 500

42. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750 A + 2A + 1/2 A = 1750

 S = 2A 7/2 A = 1750

 C = 1/2 A A = 500
S = 1000

43. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750 A + 2A + 1/2 A = 1750

 S = 2A 7/2 A = 1750

 C = 1/2 A A = 500
S = 1000
C = 250

44. Example 2
The Drama Club printed 1750 tickets for their spring play. They
printed twice as many student tickets as adult tickets and half as many
children’s tickets as adult tickets. Write a system of 3 equations and
find the number of each ticket printed.
A = adult tickets S = student tickets C = children tickets
 A + S + C = 1750 A + 2A + 1/2 A = 1750

 S = 2A 7/2 A = 1750

 C = 1/2 A A = 500
S = 1000 They printed 500 adult tickets, 1000 student
C = 250 tickets, and 250 children’s tickets

45. Example 3
Solve.
y = 4x

 xy = 36


46. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36

47. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36
4x2 = 36

48. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36
4x2 = 36
x2 = 9

49. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36
4x2 = 36
x2 = 9
x2 = ± 9

50. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36
4x2 = 36
x2 = 9
x2 = ± 9
x = 3 or x = -3

51. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3)
4x2 = 36
x2 = 9
x2 = ± 9
x = 3 or x = -3

52. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3)
4x2 = 36 y = 12
x2 = 9
x2 = ± 9
x = 3 or x = -3

53. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3)
4x2 = 36 y = 12
x2 = 9 y = 4(-3)
x2 = ± 9
x = 3 or x = -3

54. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3)
4x2 = 36 y = 12
x2 = 9 y = 4(-3)
x2 = ± 9 y = -12
x = 3 or x = -3

55. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3) Check:
4x2 = 36 y = 12
x2 = 9 y = 4(-3)
x2 = ± 9 y = -12
x = 3 or x = -3

56. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3) Check:
4x2 = 36 y = 12 (3)(12) = 36
x2 = 9 y = 4(-3)
x2 = ± 9 y = -12
x = 3 or x = -3

57. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3) Check:
4x2 = 36 y = 12 (3)(12) = 36
x2 = 9 y = 4(-3) (-3)(-12) = 36
x2 = ± 9 y = -12
x = 3 or x = -3

58. Example 3
Solve.
y = 4x

 xy = 36

x(4x) = 36 y = 4(3) Check:
4x2 = 36 y = 12 (3)(12) = 36
x2 = 9 y = 4(-3) (-3)(-12) = 36
x2 = ± 9 y = -12
x = 3 or x = -3
(3, 12) or (-3, -12)

59. Example 4
Solve.
y = 4 − 3x

3x + y = 7


60. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7

61. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4=7

62. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4≠7

63. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4≠7
Wait, what?

64. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4≠7
Wait, what?
3x + y = 7

65. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4≠7
Wait, what?
3x + y = 7
y = -3x + 7

66. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4≠7
Wait, what?
3x + y = 7
y = -3x + 7
Oh, parallel lines!

67. Example 4
Solve.
y = 4 − 3x

3x + y = 7

3x + (4 - 3x) = 7
4≠7
Wait, what?
3x + y = 7
y = -3x + 7
Oh, parallel lines!
(No solutions)

68. Example 5
Solve.
y = 2x 2

 2
3y = 6x


69. Example 5
Solve.
y = 2x 2

 2
3y = 6x

2 2
3(2x ) = 6x

70. Example 5
Solve.
y = 2x 2

 2
3y = 6x

2 2
3(2x ) = 6x
2 2
6x = 6x

71. Example 5
Solve.
y = 2x 2

 2
3y = 6x

2 2
3(2x ) = 6x
2 2
6x = 6x
This is always true!

72. Example 5
Solve.
y = 2x 2

 2
3y = 6x

2 2
3(2x ) = 6x
2 2
6x = 6x
This is always true!
These are the same graphs.

73. Example 5
Solve.
y = 2x 2

 2
3y = 6x

2 2
3(2x ) = 6x
2 2
6x = 6x
This is always true!
These are the same graphs.
Infinitely many solutions on the parabola

74. Consistent:

75. Consistent: A system with one or more solutions

76. Consistent: A system with one or more solutions
Inconsistent:

77. Consistent: A system with one or more solutions
Inconsistent: A systems with no solutions

78. Homework

79. Homework
p. 289 #1-20, skip 17, 18
“Too many people are thinking of security instead of opportunity. They
seem more afraid of life than death.” - James F. Byrnes