Question 825255
Choose any variable you want.  ..... I had a description here, but decided to delete it...


I would start with the y or the z variable in your example.  Choosing z:


Multiply the 3x+2y-z=0 by 4, and add it to the first equation.
Multiply this same equation by 3, and add it to the second equation.
You have now eliminated z, and you should have a system in x and y, without any z.

Now, decide how to multiply which equation so you can add or subtract it from the other equation and eliminate either the x or the y.  You could then simply solve for the apparant variable.  Now, find the other two variables through back-substitution for the variable you finally found.