The document provides examples for solving systems of equations using substitution. It explains the substitution method in 3 steps: 1) solve one equation for one variable, 2) substitute the expression into the other equation, and 3) solve for the variable and substitute back into the original equation. An example solves the system 4x + 3y = 27 and 2x - y = 1 by first solving the second equation for y, then substituting y = 2x - 1 into the first equation and solving for x. The solution is verified by substituting x = 3 and y = 5 back into the original equations. Another example finds the two-digit number whose digits sum to 9 and is 12 times the tens digit.

1. Section 8-3
Solve Systems by Substitution

2. Essential Question
How do you solve systems of equations using substitution?
Where you’ll see this:
Transportation, construction, sports, recreation

3. Substitution

4. Substitution
Plugging in a number or expression for a variable

5. Example 1
Solve when x = 3.
y = 3x - 2

6. Example 1
Solve when x = 3.
y = 3x - 2
y = 3(3) - 2

7. Example 1
Solve when x = 3.
y = 3x - 2
y = 3(3) - 2
y=9-2

8. Example 1
Solve when x = 3.
y = 3x - 2
y = 3(3) - 2
y=9-2
y=7

9. Solve for y
When substituting, we might need to rewrite an equation. To do
this, we need to decide which equations to rewrite.

10. Solve for y
When substituting, we might need to rewrite an equation. To do
this, we need to decide which equations to rewrite.
Which equation requires the fewest steps to solve for y?
3x −7 y = 21 8x − y =16
4x + y = 3
2 2 x+y
x +5 y = 9x −3 y =3
3 2

11. Solve a System of Equations by
Substitution

12. Solve a System of Equations by
Substitution
1. Solve one equation for one variable (your choice)

13. Solve a System of Equations by
Substitution
1. Solve one equation for one variable (your choice)
2. Substitute the expression from the equation into the other
equation

14. Solve a System of Equations by
Substitution
1. Solve one equation for one variable (your choice)
2. Substitute the expression from the equation into the other
equation
3. Solve for the variable and substitute back into the original
equation to find the other variable

15. Solve a System of Equations by
Substitution
1. Solve one equation for one variable (your choice)
2. Substitute the expression from the equation into the other
equation
3. Solve for the variable and substitute back into the original
equation to find the other variable
4. Rewrite your answer as an ordered pair and check it!

16. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1


17. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1

18. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x

19. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x
− y = −2x +1

20. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x
− y = −2x +1
y = 2x −1

21. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x
− y = −2x +1
y = 2x −1
4x +3(2x −1) = 27

22. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x
− y = −2x +1
y = 2x −1
4x +3(2x −1) = 27
4x + 6x −3 = 27

23. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x 10x −3 = 27
− y = −2x +1
y = 2x −1
4x +3(2x −1) = 27
4x + 6x −3 = 27

24. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x 10x −3 = 27
− y = −2x +1 +3 +3
y = 2x −1
4x +3(2x −1) = 27
4x + 6x −3 = 27

25. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x 10x −3 = 27
− y = −2x +1 +3 +3
y = 2x −1 10x = 30
4x +3(2x −1) = 27
4x + 6x −3 = 27

26. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x 10x −3 = 27
− y = −2x +1 +3 +3
y = 2x −1 10x = 30
10 10
4x +3(2x −1) = 27
4x + 6x −3 = 27

27. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1
−2x −2x 10x −3 = 27
− y = −2x +1 +3 +3
y = 2x −1 10x = 30
10 10
4x +3(2x −1) = 27 x =3
4x + 6x −3 = 27

28. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
− y = −2x +1 +3 +3
y = 2x −1 10x = 30
10 10
4x +3(2x −1) = 27 x =3
4x + 6x −3 = 27

29. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
+3 +3 6− y =1
− y = −2x +1
y = 2x −1 10x = 30
10 10
4x +3(2x −1) = 27 x =3
4x + 6x −3 = 27

30. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
+3 +3 6− y =1
− y = −2x +1
y = 2x −1 10x = 30 y =5
10 10
4x +3(2x −1) = 27 x =3
4x + 6x −3 = 27

31. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
+3 +3 6− y =1
− y = −2x +1
y = 2x −1 10x = 30 y =5
10 10 Check:
4x +3(2x −1) = 27 x =3
4x + 6x −3 = 27

32. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
+3 +3 6− y =1
− y = −2x +1
y = 2x −1 10x = 30 y =5
10 10 Check:
4x +3(2x −1) = 27 x =3 4(3)+3(5) = 27
4x + 6x −3 = 27

33. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
+3 +3 6− y =1
− y = −2x +1
y = 2x −1 10x = 30 y =5
10 10 Check:
4x +3(2x −1) = 27 x =3 4(3)+3(5) = 27
4x + 6x −3 = 27 2(3)−5 =1

34. Example 2
Solve the system of equations and check.
4x +3 y = 27


2x − y =1

2x − y =1 2(3)− y =1
−2x −2x 10x −3 = 27
+3 +3 6− y =1
− y = −2x +1
y = 2x −1 10x = 30 y =5
10 10 Check:
4x +3(2x −1) = 27 x =3 4(3)+3(5) = 27
4x + 6x −3 = 27 2(3)−5 =1
(3, 5)

35. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.

36. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.

37. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.
The number will be 10x + y.

38. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.
The number will be 10x + y.
Set up the system:

39. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.
The number will be 10x + y.
Set up the system:
x+ y =9

40. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.
The number will be 10x + y.
Set up the system:
x+ y =9
10x + y =12x

41. Example 3
In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.
The number will be 10x + y.
Set up the system:
x+ y =9
{ 10x + y =12x

42. Example 3
x + y = 9


10x + y =12x


43. Example 3
x + y = 9


10x + y =12x

y = −x + 9

44. Example 3
x + y = 9


10x + y =12x

y = −x + 9
10x +(−x + 9) =12x

45. Example 3
x + y = 9


10x + y =12x

y = −x + 9
10x +(−x + 9) =12x
9x + 9 =12x

46. Example 3
x + y = 9


10x + y =12x

y = −x + 9
10x +(−x + 9) =12x
9x + 9 =12x
9 = 3x

47. Example 3
x + y = 9


10x + y =12x

y = −x + 9
10x +(−x + 9) =12x
9x + 9 =12x
9 = 3x
x =3

48. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x
9x + 9 =12x
9 = 3x
x =3

49. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x y =6
9x + 9 =12x
9 = 3x
x =3

50. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x y =6
9x + 9 =12x Check:
9 = 3x
x =3

51. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x y =6
9x + 9 =12x Check:
9 = 3x 3+ 6 = 9
x =3

52. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x y =6
9x + 9 =12x Check:
9 = 3x 3+ 6 = 9
x =3
10(3)+ 6 =12(3)

53. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x y =6
9x + 9 =12x Check:
9 = 3x 3+ 6 = 9
x =3
10(3)+ 6 =12(3)
30 + 6 = 36

54. Example 3
x + y = 9


10x + y =12x

y = −x + 9 3+ y = 9
10x +(−x + 9) =12x y =6
9x + 9 =12x Check: (3, 6)
9 = 3x 3+ 6 = 9
x =3
10(3)+ 6 =12(3)
30 + 6 = 36

55. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14


56. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7

57. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7
−2(2 y +7)+ 4 y = −14

58. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7
−2(2 y +7)+ 4 y = −14
−4 y −14 + 4 y = −14

59. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7
−2(2 y +7)+ 4 y = −14
−4 y −14 + 4 y = −14
−14 = −14

60. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7
−2(2 y +7)+ 4 y = −14
−4 y −14 + 4 y = −14
−14 = −14
What’s going on here?

61. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7
−2(2 y +7)+ 4 y = −14
−4 y −14 + 4 y = −14
−14 = −14
What’s going on here?

62. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7
−2(2 y +7)+ 4 y = −14 1 7
y= x−
−4 y −14 + 4 y = −14 2 2
−14 = −14
What’s going on here?

63. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7 4 y = 2x −14
−2(2 y +7)+ 4 y = −14 1 7
y= x−
−4 y −14 + 4 y = −14 2 2
−14 = −14
What’s going on here?

64. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7 4 y = 2x −14
−2(2 y +7)+ 4 y = −14 1 7 1 7
y= x− y= x−
−4 y −14 + 4 y = −14 2 2 2 2
−14 = −14
What’s going on here?

65. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7 4 y = 2x −14
−2(2 y +7)+ 4 y = −14 1 7 1 7
y= x− y= x−
−4 y −14 + 4 y = −14 2 2 2 2
−14 = −14 These are the same lines!
What’s going on here?

66. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7 4 y = 2x −14
−2(2 y +7)+ 4 y = −14 1 7 1 7
y= x− y= x−
−4 y −14 + 4 y = −14 2 2 2 2
−14 = −14 These are the same lines!
What’s going on here? Infinitely many solutions
on the line.

67. Example 4
Solve each system of equations. Check your solution.
x − 2 y = 7

a. 
−2x + 4 y = −14

x = 2 y +7 −2 y = −x +7 4 y = 2x −14
−2(2 y +7)+ 4 y = −14 1 7 1 7
y= x− y= x−
−4 y −14 + 4 y = −14 2 2 2 2
−14 = −14 These are the same lines!
What’s going on here? Infinitely many solutions
on the line.

68. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3


69. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2

70. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2
y= x−
7 7

71. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2
y= x−
7 7
2 2
−4x +14  x −  = 3
7 7

72. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2
y= x−
7 7
2 2
−4x +14  x −  = 3
7 7
−4x + 4x − 4 = 3

73. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2
y= x−
7 7
2 2
−4x +14  x −  = 3
7 7
−4x + 4x − 4 = 3
−4 = 3

74. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2 14 y = 4x +3
y= x−
7 7
2 2
−4x +14  x −  = 3
7 7
−4x + 4x − 4 = 3
−4 = 3

75. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2 14 y = 4x +3
y= x−
7 7 2 3
2 2 y= x+
−4x +14  x −  = 3 7 14
7 7
−4x + 4x − 4 = 3
−4 = 3

76. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2 14 y = 4x +3
y= x−
7 7 2 3
2 2 y= x+
−4x +14  x −  = 3 7 14
7 7
These lines are parallel.
−4x + 4x − 4 = 3
−4 = 3

77. Example 4
Solve each system of equations. Check your solution.
2x −7 y = −2

b. 
−4x +14 y = 3

−7 y = −2x − 2
2 2 14 y = 4x +3
y= x−
7 7 2 3
2 2 y= x+
−4x +14  x −  = 3 7 14
7 7
These lines are parallel.
−4x + 4x − 4 = 3
There are no solutions.
−4 = 3

78. Homework
p. 346 #1-29 odd
“I have high expectations for you, so I will wait a little longer. I know you
can get it if I give you a chance.” - Myra and David Sadker