Systems of Equations with Three Variables
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solve 2x + y – z = 3 4x – y + 4z = 0 -3y + 2z = 6
Systems of Equations with Three Variables

1) 2x + y – z = 3
2) 4x – y + 4z = 0
3) -3y + 2z = 6

Multiply equation 1 by 2.

2 * (2x + y – z) = 3 * 2
4) 4x + 2y - 2z = 6

Subtract equation 2 from equation 4.

  4x + 2y - 2z = 6
-(4x –  y + 4z = 0)
----------------------
         3y - 6z = 6
5) 3y - 6z = 6

Add equation 3 to equation 5.

     3y - 6z = 6
+(-3y + 2z = 6)
-------------------
         -4z  = 12
-4z = 12
z = -3     <<<<<<<<<<<<<<<<<<<<

Plug that into equation 3 to solve for y.

-3y + 2z = 6
-3y + 2(-3) = 6
-3y - 6 = 6
-3y = 12
y = -4     <<<<<<<<<<<<<<<<<<<<

Plug both of those values into equation 1 to solve for x.

2x + y – z = 3
2x + (-4) – (-3) = 3
2x - 4 + 3 = 3
2x - 1 = 3
2x = 4
x = 2     <<<<<<<<<<<<<<<<<<<<

To check the answers, plug all three values into equation 2.

4x – y + 4z = 0
4(2) – (-4) + 4(-3) = 0
8 + 4 - 12 = 0
12 - 12 = 0
0 = 0

x = 2, y = -4, z = -3
by Level 11 User (78.4k points)

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