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 Question from Kim, a student: solve for x,y,z x - y + z = 4 2x + 2y - z = -2 x - 2y + 2z = 7

Hi Kim,

I will o a similar problem

solve for x,y,z

x + 2y + z = 0         (1)
2x + 5y + 4z = -1    (2)
x - y - 9z = -5          (3)

I am going to use the method of elimination and the first task is to decide which variable to eliminate first. When I look at the coefficients I see that if I subtract the first and third equation then x is eliminated so I am going to eliminate x.

Subtract equation (3) from equation (1) to get

3y + 10z = 5      (4)

Multiply equation (1) by 2 and subtract from the equation (2)

2x + 4y + 2z = 0         2 times (1)
2x + 5y + 4z = -1                    (2)
subtract
y + 2z = -1                               (5)

Multiply equation (5) by 3 and subtract from equation (4)

3y + 6z = -3                 3 times (5)
3y + 10z = 5                             (4)
subtract
4z = 8

and hence

z = 2.

Substitute z = 2 into equation (5) to get

y + 2 × 2 = -1

and thus

y = -5.

Finally substitute y = -5 and z = 2 into equation (1).

x + (2 × -5) + 2 = 0

and hence

x = 8.