Thank you for the opportunity to help you with your question!
To solve a system of equations by substitution means to replace a variable with what is equals. So, if y=z, we can replace all y's in the other equations with what y equals. All y's become "z" since y=z.
Now we have:
x+ z -3z =-1
-x+ 2z = 1.
It is best to simplify before doing the next step. Add together z and -3z.
x - 2z = -1
-x +2z =1.
Now we do the same step again. We will replace "x" with whatever "x" equals. We have to get x by itself in order to do this step.
x - 2z (+2z) = -1 (+2z)
x= -1 +2z
Now we can replace x with what x equals, in the other equation. Anytime you see an "x", put (-1 +2z) in that spot.
-x +2z =1
-- (-1 +2z) +2z =1 (HINT: do not forget about the negative that was in front of x. We have to distribute)
1 -2z +2z = 1
1=1 WHEN THE VARIABLE DISAPPEARS IN THE ANSWER AND BOTH SIDES EQUAL EACH OTHER, THE ANSWER IS "ALL REAL NUMBERS".
so x, y, and z = all real answers
Please let me know if you need any clarification. I'm always happy to answer your questions.