Systems of Equations in Three Variables
2x+y-z=-8
4x-y+2z=-3
-3x+y+2z=5
1) 2x + y - z = -8
2) 4x - y + 2z = -3
3) -3x + y + 2z = 5
Multiply equation 1 by 2.
2(2x + y - z) = -8 * 2
4) 4x + 2y - 2z = -16
Add equation 2 to equation 4.
4x + 2y - 2z = -16
+(4x - y + 2z = -3)
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8x + y = -19
5) 8x + y = -19
Subtract equation 3 from equation 2.
4x - y + 2z = -3
-(-3x + y + 2z = 5)
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7x - 2y = -8
6) 7x - 2y = -8
Multiply equation 5 by 2.
2(8x + y) = -19 * 2
7) 16x + 2y = -38
Add equation 6 to equation 7.
16x + 2y = -38
+(7x - 2y = -8)
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23x = -46
23x = -46
x = -2 <<<<<<<<<<<<<<<<<
Use equation 5 to solve for y.
8x + y = -19
8(-2) + y = -19
-16 + y = -19
y = -3 <<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
-3x + y + 2z = 5
-3(-2) + (-3) + 2z = 5
6 - 3 + 2z = 5
3 + 2z = 5
2z = 5 - 3
2z = 2
z = 1 <<<<<<<<<<<<<<<<<
Check, using the original equations.
1) 2x + y - z = -8
2(-2) + (-3) - 1 = -8
-4 - 3 - 1 = -8
-8 = -8
2) 4x - y + 2z = -3
4(-2) - (-3) + 2(1) = -3
-8 + 3 + 2 = -3
-3 = -3
3) -3x + y + 2z = 5
-3(-2) + (-3) + 2(1) = 5
6 - 3 + 2 = 5
5 = 5
All good.
Answer: x = -2, y = -3, z = 1