x + y + z = 8
2x + 5y + 2z = 25
-x + 6y - 3z = 7
add the first to the third
x + y + z = 8
2x + 5y + 2z = 25
0x + 7y - 2z = 15
multiply the first by -2 on both sides
-2x - 2y - 2z = -16
2x + 5y + 2z = 25
0x + 7y - 2z = 15
add the second to the first
0x + 3y + 0z = 9
2x + 5y + 2z = 25
0x + 7y - 2z = 25
divide the first by 3 on both sides
0x + y + 0z = 3
2x + 5y + 2z = 25
0x + 7y - 2z = 25
Now we know y = 3
multiply the first by -7 on both sides
0x - 7y + 0z = -21
2x + 5y + 2z = 25
0x + 7y - 2z = 25
add the first to the third
0x - 7y + 0z = -21
2x + 5y + 2z = 25
0x + 0y - 2z = 4
divide the third by -2
0x - 7y + 0z = -21
2x + 5y + 2z = 25
0x + 0y + z = -2
No we know z = -2
divide the first by -7
0x + y + 0z = 3
2x + 5y + 2z = 25
0x + 0y + z = -2
multiply the first by -5 and the third by -2
0x - 5y + 0z = -15
2x + 5y + 2z = 25
0x + 0y - 2z = 4
add the first and third to the second
0x - 5y + 0z = -15
2x + 0y + 0z = 14
0x + 0y - 2z = 4
divide the second by 2
0x - 5y + 0z = -15
x + 0y + 0z = 7
0x + 0y - 2z = 4
Now we know x = 7
Answer: (7, 3, -2)