Line the equations up so that the letters line up vertically,
like this:
6x + z = 21
4x + 6y = 34
-5y + 6z = -33
Notice that y is already eliminated from the first equation,
z is already eliminated from the second equation and x is
already eliminated from the third equation. We need two
equations with the SAME letter eliminated from them. So we
pick the easiest two equations to eliminate a letter from.
We pick the 1st and 3rd to eliminate z from:
6x + z = 21
-5y + 6z = -33
We multiply the top equation through by -6 to make it cancel
with the bottom equation, Then we add vertically term-by-term
-36x + -6z = -126
-5y + 6z = -33
----------------------
-36x - 5y = -159
Since we have eliminated z, and the original second equation also
has z eliminated we put those two together:
4x + 6y = 34
-36x - 5y = -159
-----------------
We can make the x's cancel by multiplying the top equation by 9
36x + 54y = 306
-36x - 5y = -159
------------------
49y = 147
y = 147/49
y = 3
Substitute y = 3 into
4x + 6y = 34
4x + 6(3) = 34
4x + 18 = 34
4x = 16
x = 4
Substitute x = 4 into the original first equation:
6x + z = 21
6(4) + z = 21
24 + z = 21
z = -3
So the solution is (x, y, z) = (4, 3, -3)
Edwin