Problem: x+y+z=6 and 3x-y+2z=7 and 3y-4z=-6
Simultaneous equation
1) x + y + z = 6
2) 3x - y + 2z = 7
3) 3y - 4z = -6
Multiply equation 1 by 2.
2(x + y + z) = 6 * 2
4) 2x + 2y + 2z = 12
Subtract equation 2 from equation 4.
2x + 2y + 2z = 12
-(3x - y + 2z = 7)
-------------------------
-x + 3y = 5
5) -x + 3y = 5
Multiply equation 2 by 4.
4(3x - y + 2z) = 7 * 4
6) 12x - 4y + 8z = 28
Multiply equation 3 by 2.
2(3y - 4z) = -6 * 2
7) 6y - 8z = -12
Add equation 7 to equation 6.
12x - 4y + 8z = 28
+ (6y - 8z = -12)
--------------------------
12x + 2y = 16
8) 12x + 2y = 16
Multiply equation 5 by 2.
2(-x + 3y) = 5 * 2
9) -2x + 6y = 10
Multiply equation 8 by 3.
3(12x + 2y) = 16 * 3
10) 36x + 6y = 48
Subtract equation 10 from equation 9.
-2x + 6y = 10
-(36x + 6y = 48)
----------------------
-38x = -38
-38x = -38
x = 1 <<<<<<<<<<<<<<<<
Use equation 5 to solve for y.
-x + 3y = 5
-1 + 3y = 5
3y = 6
y = 2 <<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
3y - 4z = -6
3(2) - 4z = -6
6 - 4z = -6
-4z = -12
z = 3 <<<<<<<<<<<<<<<<
Check....
1) x + y + z = 6
1 + 2 + 3 = 6
6 = 6
2) 3x - y + 2z = 7
3(1) - 2 + 2(3) = 7
3 - 2 + 6 = 7
7 = 7
3) 3y - 4z = -6
3(2) - 4(3) = -6
6 - 12 = -6
-6 = -6
Answer: x = 1, y = 2, z = 3