(1)
[1] x + 5y - 4z = -10
[2] 2x - y + 5z = -9
[3] 2x - 10y - 5z = 0
Eliminate x by adding -2[1] to [2], and adding -1[2] to [3]:
-2(x + 5y - 4z) + (2x - y + 5z) = -2(-10) + (-9)
-2x - 10y + 8z + 2x - y + 5z = 20 - 9
[4] -11y + 13z = 11
-(2x - y + 5z) + (2x - 10y - 5z) = -(-9) + 0
-2x + y - 5z + 2x - 10y - 5z = 9
[5] -9y - 10z = 9
Eliminate y by adding -9[4] to 11[5]:
-9(-11y + 13z) + 11(-9y - 10z) = -9(11) + 11(9)
99y - 117z - 99y - 110z = -99 + 99
-227z = 0
z = 0
Plug this solution into either [4] or [5] and solve for y :
[5] -9y - 10(0) = 9
-9y = 9
y = -1
Plug them both into either of [1], [2], or [3] and solve for x :
[1] x + 5(-1) - 4(0) = -10
x - 5 = -10
x = -5
(2)
[1] 2x - y + z = -4
[2] z = 5
[3] -2x + 3y - z = -10
Substitute [2] into [1] and [3]:
2x - y + 5 = -4
[4] 2x - y = -9
-2x + 3y - 5 = -10
[5] -2x + 3y = -5
Solve [4] for y :
2x - y = -9
2x + 9 = y
Substitute this into [5] and solve for x :
-2x + 3(2x + 9) = -5
-2x + 6x + 27 = -5
4x = -32
x = -8
Plug this into either [4] or [5] to solve for y :
[4] 2(-8) - y = -9
-16 - y = -9
y = -5
