-x-3z=-2 3x+y-2z=5 2x+2y+z=4
1) -x - 3z = -2
2) 3x + y - 2z = 5
3) 2x + 2y + z = 4
Multiply equation 2 by 2.
2 * (3x + y - 2z) = 5 * 2
4) 6x + 2y - 4z = 10
Subtract equation 3 from equation 4.
6x + 2y - 4z = 10
-(2x + 2y + z = 4)
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4x - 5z = 6
5) 4x - 5z = 6
Multiply equation 1 by 4.
4 * (-x - 3z) = -2 * 4
6) -4x - 12z = -8
Add equation 5 to equation 6.
-4x - 12z = -8
+(4x - 5z = 6)
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- 17z = -2
-17z = -2
z = 2/17 <<<<<<<<<<<<<<<<<<<<<<
Plug that into equation 1 to solve for x.
-x - 3z = -2
-x - 3(2/17) = -2
-x - 6/17 = -2
-x = -2 + 6/17
-x = -1 11/17
x = 1 11/17 <<<<<<<<<<<<<<<<<<<<<<
It will be easier to use that as an improper fraction: 28/17
Plug the values for x and z into equation 2 and solve for y.
3x + y - 2z = 5
3(28/17) + y - 2(2/17) = 5
84/17 + y - 4/17 = 5
80/17 + y = 5
y = 5 - 80/17
y = 85/17 - 80/17
y = 5/17 <<<<<<<<<<<<<<<<<<<<<<
We'll use equation 3 to check the answer
2x + 2y + z = 4
2(28/17) + 2(5/17) + 2/17 = 4
56/17 + 10/17 + 2/17 = 4
68/17 = 4
4 = 4
We can use equation 2, also.
3x + y - 2z = 5
3(28/17) + 5/17 - 2(2/17) = 5
84/17 + 5/17 - 4/17 = 5
85/17 = 5
5 = 5
They check.
x = 28/17, y = 5/17, z = 2/17