1. 3.6 Systems with Three Variables
Solving Three-Variable Systems by
Substitution

2. 2) Solving Three-Variable Systems by Substitution
Substitution may be easier in cases where an equation can be solved for one variable
Example: x + 32y – 32.7z = -382
Rewrite as: x = -382 – 32y z

3. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10 {

4. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Re-write (1) as x =. { 1 2 3

5. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Re-write (1) as x =.
x = y – z { 1 2 3 1

6. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Now…it gets ugly. Sub x = y – z in { 1 2 3 2

7. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Sub x = y – z in { 1 2 3 2 2

8. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Sub x = y – z in
-4(-4 + 2y – z) + y – 2z = 1 { 1 2 3 2 2

9. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Sub x = y – z in
-4(-4 + 2y – z) + y – 2z = 1
16 – 8y + 4z + y – 2z = 1
-7y + 2z = -15 { 1 2 3 2 2 4

10. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Sub x = y – z in { 1 2 3 3 3

11. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Sub x = y – z
2(-4 + 2y – z) + 2y – z = 10 { 1 2 3 3 3

12. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Sub x = y – z
2(-4 + 2y – z) + 2y – z = 10
-8 + 4y – 2z + 2y – z = 10
6y – 3z = 18 { 1 2 3 3 3 5

13. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Solve for y and z with elimination, substitution.
-7y + 2z = -15
6y – 3z = 18 { 1 2 3 4 5

14. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Solve for y and z with elimination, substitution.
-7y + 2z = -15 x 3
6y – 3z = 18 x 2 { 1 2 3 4 5

15. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Solve for y and z with elimination, substitution.
-7y + 2z = -15 x y + 6z = -45
6y – 3z = 18 x y – 6z = 36 { 1 2 3 4 5

16. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Solve for y and z with elimination, substitution.
-7y + 2z = -15 x y + 6z = -45
6y – 3z = 18 x y – 6z = 36
-9y = -9
y = 1 { 1 2 3 4 5

17. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Solve for y and z with elimination, substitution.
-7y + 2z = -15
-7(1) + 2z = -15
2z = z = -4 { 1 2 3 4 4
Sub y = 1 in

18. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Solve for x.
x = y – z
x = (1) –(-4)
x =
x = 2 { 1 2 3
Sub y = 1, z = -4.

19. 2) Solving Three-Variable Systems by Substitution
Example 1:
Solve the system by substitution x – 2y + z = -4
-4x + y – 2z = 1
2x + 2y – z = 10
Therefore, the solution is (2, 1, -4). { 1 2 3

20. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12 {

21. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange Sub in and { 1 2 3 2 1 3

22. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange Sub in and
y = x + 14z { 1 2 3 2 1 3 2

23. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange Sub in and
y = x + 14z x + 7y + 5z = 16 { 1 2 3 2 1 3 2 1

24. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange Sub in and
y = x + 14z x + 7y + 5z = 16
12x + 7(-9 + 2x + 14z) + 5z = 16
12x x + 98z + 5z = 16
26x + 103z = 79 { 1 2 3 2 1 3 2 1 4

25. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange Sub in and
y = x + 14z x - 2y + 9z = -12
-3x - 2(-9 + 2x + 14z) + 9z = -12 { 1 2 3 2 1 3 2 3

26. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange Sub in and
y = x + 14z x - 2y + 9z = -12
-3x - 2(-9 + 2x + 14z) + 9z = -12
-3x + 18 – 4x – 28z + 9z = -12
-7x - 19z = -30 { 1 2 3 2 1 3 2 3 5

27. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Add and
26x + 103z = 79
-7x - 19z = -30 { 1 2 3 4 5 4 5

28. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Add and
26x + 103z = 79 x x + 721z = 553
-7x - 19z = -30 x x – 494z = -780 { 1 2 3 4 5 4 5

29. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Add and
26x + 103z = 79 x x + 721z = 553
-7x - 19z = -30 x x – 494z = -780
227z = -227
z = -1 { 1 2 3 4 5 4 5

30. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Sub z = -1 in
26x + 103z = 79
26x + 103(-1) = 79
x = 7 { 1 2 3 4

31. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Sub z = -1 and x = 7 in
12x + 7y + 5z = 16
12(7) + 7y + 5(-1) = 16
84 + 7y – 5 = 16
y = -9 { 1 2 3 1

32. 2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Therefore, the solution is (7, -9, -1). { 1 2 3

33. Homework
p.157 #10-12, 28, 29, 38, 39
ASSIGNMENT: 3.5, 3.6 on Monday Oct 19
QUIZ: 3.5, 3.6 on Wednesday Oct 21
TEST: 3.1, 3.2, 3.5, 3.6 on Monday Oct 26
Halloween Party: Friday October 30
5-8pm
Dancing, food and games