2x + 5y - 3z = -5
3x + 2y + 7z = 15
5x - 4y + 6z = 34
multiply the top equation by 2 on both sides
4x + 10y - 6z = -10
3x + 2y + 7z = 15
5x - 4y + 6z = 34
add the bottom equation to the top equation
9x + 6y + 0z = 24
3x + 2y + 7z = 15
5x - 4y + 6z = 34
multiply the middle equation by 6 on both sides and the bottom equation by -7 on both sides
9x + 6y + 0z = 24
18x + 12y + 42z = 90
-35x + 28y - 42z = -238
add the bottom equation to the middle equation.
9x + 6y + 0z = 24
-17x + 40y + 0z = -148
-35x + 28y - 42z = -238
mutiply the top equation by 20 on both sides and the middle equation by -3 on both sides
180x + 120y + 0z = 480
51x - 120y + 0z = 444
-35x + 28y - 42z = -238
add the middle equation to the top equation
231x + 0y + 0z = 924
51x - 120y + 0z = 444
-35x + 28y - 42z = -238
divide the top equation by 231 on both sides
x + 0y + 0z = 924/231
51x - 120y + 0z = 444
-35x + 28y - 42z = -238
reduce 924/231
x + 0y + 0z = 4
51x - 120y + 0z = 444
-35x + 28y - 42z = -238
Now we know x = 4
Let's plug x = 4 into the middle equation:
204 - 120y + 0z = 444
-120y = 220
y = -220/120
y = -11/6
Now we know y = -11/6
Let's plug x = 4 and y = -11/6 into the bottom equation:
-140 - 308/6 - 42z = -238
-140 - 154/3 - 42z = -238
-520 - 154 - 126z = -714
-674 - 126z = -714
126z = -40
z = -40/126
z = -20/63
Answer: (4, -11/6, -20/63)