How do i solve this using the elimination method

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.