Problem: Solve the system: 2x+y-2z=-1; 3x-3y-z=5; x-2y+3z=6
1) 2x + y - 2z = -1
2) 3x - 3y - z = 5
3) x - 2y + 3z = 6
Multiply equation 2 by 2.
2(3x - 3y - z) = 5 * 2
4) 6x - 6y - 2z = 10
Subtract equation 4 from equation 1.
2x + y - 2z = -1
-(6x - 6y - 2z = 10)
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-4x + 7y = -11
5) -4x + 7y = -11
Multiply equation 2 by 3.
3(3x - 3y - z) = 5 * 3
6) 9x - 9y - 3z = 15
Add equation 3 to equation 6.
9x - 9y - 3z = 15
+( x - 2y + 3z = 6)
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10x - 11y = 21
7) 10x - 11y = 21
Multiply equation 5 by 11.
11(-4x + 7y) = -11 * 11
8) -44x + 77y = -121
Multiply equation 7 by 7.
7(10x - 11y) = 21 * 7
9) 70x - 77y = 147
Add equation 9 to equation 8.
-44x + 77y = -121
+(70x - 77y = 147)
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26x = 26
26x = 16
x = 1 <<<<<<<<<<<<<<<<<<<
Use equation 5 to solve for y.
-4x + 7y = -11
-4(1) + 7y = -11
-4 + 7y = -11
7y = -7
y = -1 <<<<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
x - 2y + 3z = 6
1 - 2(-1) + 3z = 6
1 + 2 + 3z = 6
3 + 3z = 6
3z = 3
x = 1 <<<<<<<<<<<<<<<<<<<
Check.
1) 2x + y - 2z = -1
2(1) + (-1) - 2(1) = -1
2 - 1 -2 = -1
-1 = -1
2) 3x - 3y - z = 5
3(1) - 3(-1) - 1 = 5
3 + 3 - 1 = 5
5 = 5
Answer: x = 1, y = -1, z = 1