linear equation with three variables

[cshore480]

Need to find x, y and z in the three linear equations

x + y - 2z = 5
x + 2y + z = 8
2x + 3y -z = 13

first I multplied the first equation by -2 and added it to the second equation as follows to solve for x:

-2x - 2y + 4z = -10
x + 2y + z = 8

my new equation is
-x + 5z = -2

Therefore X = 2 + 5z

Next I multiplied the first equation by -2 and added it to the third equation to solve for y as follows:

-2x - 2y + 4z = -10
2x + 3y - z = 13

My new equation is y + 3z = 3

Therefore y = 3 - 3z

I can't figure out how to solve for Z. Substitution didn't seem to work nor did adding any of the equations together.

 

[BigGlenntheHeavy]

\(\displaystyle Let \ z \ = \ t, \then \ x \ = \ 2+5t \ and \ y \ = \ 3-3t, \ t \ a \ parameter.\)

\(\displaystyle Now. for \ example \ let \ z \ = \ t \ = \ 7. \ Then \ x \ = \ 37 \ and \ y \ = \ -18.\)

\(\displaystyle I'll \ leave \ you \ to \ do \ the \ check.\)