Essentially, you multiply one row by a number and add it to another to elimate a variable.
Multiplying the first equation by -2 and adding it to the second, we get
-3y+4z=-13
Multiplying the first equation by -4 (because x*-4+4x=0) and adding it to the third, we get
-5y+13z=-28
We have 2 new equations -
-3y+4z=-13
-5y+13z=-28
Multiplying the first new equation by -5/3 and adding it to the second new one, we get -20z/3+13z=-65/3+28
solving for z, we get (-65/3+28)/(-20/3+13)=1. After that, it's easy to solve for y and x.