Solve a system of equations with three variables using elimination
Learn how to solve a system of equations with three variables with one solution using elimination method. We will number the equations in order to make the procedure easy to follow.
Example #1:
1. x - 3y + 3z = -6
2. 2x + 3y - z = 15
3. 4x - 3y - z = 21
The goal is to pick a pair of equations and eliminate a variable by adding the two equations. And then pick another pair of equations and eliminate the exact same variable. Once you have done that, you will be left with a pair of equations with just 2 variables.
Step 1
Notice that you can easily eliminate y because the y terms are already additive inverse.
Therefore, pair 1. with 2. and 2. with 3. so you can easily get rid of the y terms.
Add the left sides of 1. and 2. and add the right sides of 1. and 2.
You get 3x + 2z = 9 and we call this new equation 4.
1. x - 3y + 3z = -6
2. 2x + 3y - z = 15
________________
4. 3x + 2z = 9
Add the left sides of 2. and 3. and add the right sides of 2. and 3.
You get 6x - 2z = 36 and we call this new equation 5.
2. 2x + 3y - z = 15
3. 4x - 3y - z = 21
_________________
5. 6x - 2z = 36
Step 2
Write the two new equations 4. and 5. as a system.
4. 3x + 2z = 9
5. 6x - 2z = 36
Solve for x and z.
Since the z terms are already additive inverses, you can just add the left sides of 4. and 5. and add the right sides of 4. and 5.
4. 3x + 2z = 9
5. 6x - 2z = 36
_________________
9x = 45
x = 5
Use either equation 4. or 5. to find z
4. 3x + 2z = 9
3(5) + 2z = 9
15 + 2z = 9
2z = 9 - 15
2z = -6
z = -3
Step 3
Use either equation 1. 2. or 3. to find y.
2. 2x + 3y - z = 15
2(5) + 3y - -3 = 15
10 + 3y + 3 = 15
13 + 3y = 15
3y = 15 - 13
3y = 2
y = 2/3
The solution of the system is (5, 2/3, -3)
Example #2:
1. 6x - y + 2z = 8
2. 2x + 3y - z = -9
3. 4x + 2y + 5z = 1
Step 1
Multiply equation 2. by 2.
2. 2(2x + 3y - z) = 2(-9)
2. 4x + 6y - 2z = -18
Looking at 1. and 2. you can easily eliminate z because the z terms are now additive inverse.
1. 6x - y + 2z = 8
2. 4x + 6y - 2z = -18
Add the left sides of 1. and 2. and add the right sides of 1. and 2.
You get 10x + 5y = -10 and we call this new equation 4.
1. 6x - y + 2z = 8
2. 4x + 6y - 2z = -18
__________________
4. 10x + 5y = -10
Multiply equation 2. by 5.
2. 5(2x + 3y - z) = 5(-9)
2. 10x + 15y - 5z = -45
Looking at 3. and 2. you can easily eliminate z because the z terms are now additive inverse.
2. 10x + 15y - 5z = -45
3. 4x + 2y + 5z = 1
Add the left sides of 2. and 3. and add the right sides of 2. and 3.
You get 14x + 17y = -44 and we call this new equation 5.
2. 10x + 15y - 5z = -45
3. 4x + 2y + 5z = 1
____________________
5. 14x + 17y = -44
Step 2
Write the two new equations 4. and 5. as a system.
4. 10x + 5y = -10
5. 14x + 17y = -44
Solve for x and y.
Multiply 4. by -14 and 5. by 10
4. -14(10x + 5y) = -14(-10)
5. 10(14x + 17y) = 10(-44)
4. -140x + -70y = 140
5. 140x + 170y = -440
Since the x terms are now additive inverses, you can just add the left sides of 4. and 5. and add the right sides of 4. and 5.
4. -140x + -70y = 140
5. 140x + 170y = -440
______________________
100y = -300
y = -3
Use either equation 4. or 5. to find x
4. 10x + 5y = -10
10x + 5(-3) = -10
10x + -15 = -10
10x = 5
x = 5/10
x = 1/2
Use either equation 1. 2. or 3. to find z.
1. 6x - y + 2z = 8
6(1/2) - -3 + 2z = 8
3 + 3 + 2z = 8
6 + 2z = 8
2z = 2
z = 1
The solution of the system is (1/2, -3, 1)
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