What is the solution to the system?
5x-y+z=4
x+2y-z=5
2x+3y-3z=5

Eliminate one variable at a time.  We have three equations so we can solve for three variables.

(5x-y+z=4)-5(x+2y-z=5)=-11y+6z=-21

-2(x+2y-z=5)+(2x+3y-3z=5)=-y-z=-5

Now using the two yz equations above to cancel out z

(-11y+6z=-21)+6(-y-z=-5)=-17y=-51

-17y=-51  divide both sides by -17

y=3, making -y-z=-5 become:

-3-z=-5, z=2, making 5x-y+z=4 become:

5x-3+2=4

5x-1=4

5x=5

x=1

So the solution to the system of equations, (x,y,z) is (1,3,2)