3.6 systems with three variables 2.solving three-variable systems by substitution
TRANSCRIPT
3.6 Systems with Three Variables
2. Solving Three-Variable Systems bySubstitution
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 + 32.7z
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{
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 =.
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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 = -4 + 2y – z
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1
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 = -4 + 2y – z in
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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 = -4 + 2y – z in -4x + y – 2z = 1
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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 = -4 + 2y – z in -4x + y – 2z = 1
-4(-4 + 2y – z) + y – 2z = 1
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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 = -4 + 2y – z in -4x + y – 2z = 1
-4(-4 + 2y – z) + y – 2z = 1
16 – 8y + 4z + y – 2z = 1
-7y + 2z = -15
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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 = -4 + 2y – z in
2x + 2y – z = 10
{1
2
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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
Sub x = -4 + 2y – z
2x + 2y – z = 10
2(-4 + 2y – z) + 2y – z = 10
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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 = -4 + 2y – z
2x + 2y – z = 10
2(-4 + 2y – z) + 2y – z = 10
-8 + 4y – 2z + 2y – z = 10
6y – 3z = 18
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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
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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
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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 -21y + 6z = -45
6y – 3z = 18 x 2 12y – 6z = 36
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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 -21y + 6z = -45
6y – 3z = 18 x 2 12y – 6z = 36
-9y = -9
y = 1
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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 = -8 z = -4
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Sub y = 1 in 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
Solve for x.
x = -4 + 2y – z
x = -4 + 2(1) –(-4)
x = -4 + 2 + 4
x = 2
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Sub y = 1, z = -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
Therefore, the solution is (2, 1, -4).
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
{
2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange . Sub in and .
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12 3
2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange . Sub in and .
y = -9 + 2x + 14z
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange . Sub in and .
y = -9 + 2x + 14z 12x + 7y + 5z = 16
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12 3
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange . Sub in and .
y = -9 + 2x + 14z 12x + 7y + 5z = 16
12x + 7(-9 + 2x + 14z) + 5z = 16
12x - 63 + 14x + 98z + 5z = 16
26x + 103z = 79
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange . Sub in and .
y = -9 + 2x + 14z -3x - 2y + 9z = -12
-3x - 2(-9 + 2x + 14z) + 9z = -12
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12 3
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Rearrange . Sub in and .
y = -9 + 2x + 14z -3x - 2y + 9z = -12
-3x - 2(-9 + 2x + 14z) + 9z = -12
-3x + 18 – 4x – 28z + 9z = -12
-7x - 19z = -30
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Add and .
26x + 103z = 79
-7x - 19z = -30
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54
2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Add and .
26x + 103z = 79 x 7 182x + 721z = 553
-7x - 19z = -30 x 26 -182x – 494z = -780
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Add and .
26x + 103z = 79 x 7 182x + 721z = 553
-7x - 19z = -30 x 26 -182x – 494z = -780
227z = -227
z = -1
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Sub z = -1 in
26x + 103z = 79
26x + 103(-1) = 79
x = 7
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2) Solving Three-Variable Systems by Substitution
Example 2:Solve the system by substitution. 12x + 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
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2) Solving Three-Variable Systems by Substitution
Example 2:
Solve the system by substitution. 12x + 7y + 5z = 16
-2x + y – 14z = -9
-3x - 2y + 9z = -12
Therefore, the solution is (7, -9, -1).
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Homework
p.157 #10-12, 28, 29, 38, 39
ASSIGNMENT: 3.5, 3.6 on Monday Oct 19QUIZ: 3.5, 3.6 on Wednesday Oct 21TEST: 3.1, 3.2, 3.5, 3.6 on Monday Oct 26
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