The problem is:
2x + 5y + 4z = 21
4x - 5y +z = 38
6x - z = 17
step by step please.
Usually I would say to eliminate the x values first, but in this case it looks easier to eliminate the z's.
Do the following.
$\displaystyle R2: R_2 + R_3$ and $\displaystyle R_1: R1 + 4R_3$
$\displaystyle 26x + 5y = 87$
$\displaystyle 10x - 5y = 55$
$\displaystyle 6x - z = 17$
Now do $\displaystyle R_1: R_1 + R_2$
$\displaystyle 36x = 144$
$\displaystyle 10x - 5y = 55$
$\displaystyle 6x - z = 17$.
Now you can solve for x. $\displaystyle x = 4$.
Substitute this value into the other two equations to solve for y and z.
$\displaystyle 40 - 5y = 55$
$\displaystyle -5y = 15$
$\displaystyle y = -3$.
$\displaystyle 24 - z = 17$
$\displaystyle z = 7$.
So your solution is [tex](x, y, z) = (4, -3, 7).
You could substitute these back into your original equation to check if you wish.