[1] 4x - 2y + 3z = 1
[2] x + 3y - 4z = -7
[3] 3x + y + 2z = 5
Add -4 [2] to [1], and add -3 [2] to [3] to eliminate x :
-4 (x + 3y - 4z) + (4x - 2y + 3z) = -4 (-7) + 1
-4x - 12y + 16z + 4x - 2y + 3z = 28 + 1
[4] -14y + 19z = 29
-3(x + 3y - 4z) + (3x + y + 2z) = -3 (-7) + 5
-3x - 9y + 12z + 3x + y + 2z = 21 + 5
-8y + 14z = 26
[5] -4y + 7z = 13
Add 2 [4] to -7 [5] to eliminate y :
2 (-14y + 19z) - 7 (-4y + 7z) = 2 (29) - 7 (13)
-28y + 38z + 28y - 49z = 58 - 91
-11z = -33
z = 3
Plug this into either [4] or [5] to solve for y :
-4y + 7 (3) = 13
-4y + 21 = 13
-4y = -8
y = 2
Plug both of these solutions into either of [1], [2], or [3] to solve for x :
x + 3 (2) - 4 (3) = -7
x + 6 - 12 = -7
x = -1