Three Variable Systems
x+y+z=-8
2x+5y+2z=-34
-x+7y-3z=-38
1) x + y + z = -8
2) 2x + 5y + 2z = -34
3) -x + 7y - 3z = -38
Multiply equation 1 by 2.
2(x + y + z) = -8 * 2
4) 2x + 2y + 2z = -16
Subtract equation 2 from equation 4.
2x + 2y + 2z = -16
-(2x + 5y + 2z = -34)
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- 3y = 18
5) -3y = 18
y = -6 <<<<<<<<<<<<<<<<<
Multiply equation equation 2 by 3.
3(2x + 5y + 2z) = -34 * 3
6) 6x + 15y + 6z = -102
Multiply equatin 3 by 2.
2(-x + 7y - 3z) = -38 * 2
7) -2x + 14y - 6z = -76
Add equation 7 to equation 6.
6x + 15y + 6z = -102
+(-2x + 14y - 6z = -76)
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4x + 29y = -178
8) 4x + 29y = -178
Multiply equation 5 by 29.
29(-3y) = 18 * 29
9) -87y = 522
Multiply equation 8 by 3.
3(4x + 29y) = -178 * 3
10) 12x + 87y = -534
Add equation 10 to equation 9.
-87y = 522
+(12x + 87y = -534)
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12x = -12
12x = -12
x = -1 <<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
-x + 7y - 3z = -38
-(-1) + 7(-6) - 3z = -38
1 - 42 - 3z = -38
-41 - 3z = -38
-3z = -38 + 41
-3z = 3
z = -1 <<<<<<<<<<<<<<<<<
Check the answers.
1) x + y + z = -8
(-1) + (-6) + (-1) = -8
-1 - 6 - 1 = -8
-8 = -8
2) 2x + 5y + 2z = -34
2(-1) + 5(-6) + 2(-1) = -34
-2 -30 - 2 = -34
-34 = -34
3) -x + 7y - 3z = -38
-(-1) + 7(-6) - 3(-1) = -38
1 - 42 + 3 = -38
-38 = -38
Everything checks out.
Answer: x = -1, y = -6, z = -1