X+y+z=9

2x-2y+4z = 0

find the solution set of the linear system. write solution using a parameter

x=-3y+18

z=2y-9.

Step-by-step explanation:

x+y+z=9, which means that x=9-y-z.

2x-2y+4z=0, which, by dividing both sides by 2, means that x-y+2z=0.

Substituting x from the first formula we have (9-y-z) - y+2z=0, which means that z=2y-9. Using the formula for x we have x=9-y-z=9-y - (2y-9) = -3y+18.

So our solutions, using y as a parameter, are x=-3y+18 and z=2y-9.

We can verify if this is true (and no mistakes were done in the derivation) by substituting in the original formulas:

x+y+z = (-3y+18) + y + (2y-9) = 9, which we know must be true, and

2 (-3y+18) - 2y+4 (2y-9) = -6y+36-2y+8y-36=0, which we know must be true.