Solve the following system of equations:

START this way:

Multiply FIRST equation by  -2

   then add to equation 3 

      this will allow you to solve for z...........continue on your own and see if you can find x and y

First....there is no "set" way of doing this.....using EP's  idea we have 

-2x + 2y - 6z  = -12

2x -  2y + 5z  = 9          add these  and we get

-z  = -3

z = 3

Now using the first equation we have that 

x - y +  3(3) = 6

x - y + 9  = 6        subtract 9 from both sides

x - y  = -3    

So now we have two equations to work with

x - y  = - 3     multiply through by - 1   ⇒   -x + y  = 3      (4)

x - 2y  = 5       (5)

Add  (4)  and (5)  and we have that

-y = 8   ⇒    y  = - 8

And using (4) to find x   -x + (-8) = 3       add 8 to both sides

-x = 11

x = -11

So

{x, y, z}  = { -11, -8, 3 }

These take some real practice, GM.......also......easy to make mistakes.....be careful !!!