Question

Solve the following system of equations for all three variables.

Answer 1

Eliminate z by combining the first two equations:

(7x + 9y - 2z) - 2 (x - 9y - z) = 9 - 2 (-3)

7x + 9y - 2z - 2x + 18y + 2z = 9 + 6

5x + 27y = 15

Eliminate z by combining the last two equations:

5 (x - 9y - z) + (-8x - 9y + 5z) = 5 (-3) + 6

5x - 45y - 5z - 8x - 9y + 5z = -15 + 6

-3x - 54y = -9

x + 18y = 3

Now eliminate x from these two new equations and solve for y :

(5x + 27y) - 5 (x + 18y) = 15 - 5 (3)

5x + 27y - 5x - 90y = 15 - 15

-63y = 0

y = 0

It follows that

x + 18y = x = 3

and

x - 9y - z = 3 - z = -3   ⇒   -z = -6   ⇒   z = 6