Problem: solve system equations
x-4y-2z=1
2x-5y+8z=16
x-3y+2z=5
1) x - 4y - 2z = 1
2) 2x - 5y + 8z = 16
3) x - 3y + 2z = 5
Multiply equation 1 by 8.
8(x - 4y - 2z) = 1 * 8
4) 8x - 32y - 16z = 8
Multiply equation 2 by 2.
2(2x - 5y + 8z) = 16 * 2
5) 4x + 10y + 16z = 32
Add equation 5 to equation 4.
8x - 32y - 16z = 8
+(4x + 10y + 16z = 32)
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12x - 22y = 40
6) 12x - 22y = 40
Multiply equation 3 by 8.
8(x - 3y + 2z) = 5 * 8
7) 8x - 24y + 16z = 40
Subtract equation 7 from equation 5.
4x + 10y + 16z = 32
-(8x - 24y + 16z = 40)
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-4x + 34y = -8
8) -4x + 34y = -8
Multiply equation 6 by 17.
17(12x - 22y) = 40 * 17
9) 204x - 374y = 680
Multiply equation 8 by 11.
11(-4x + 34y) = -8 * 11
10) -44x + 374y = -88
Add equation 10 to equation 9
204x - 374y = 680
+(-44x + 374y = -88)
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160x = 592
160x = 592
x = 3.7 <<<<<<<<<<<<<<<<<<
Use equation 6 to solve for y.
12x - 22y = 40
12(3.7) - 22y = 40
44.4 - 22y = 40
-22y = -4.4
y = 0.2 <<<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
x - 3y + 2z = 5
3.7 - 3(0.2) + 2z = 5
3.7 - 0.6 + 2z = 5
3.1 + 2z = 5
2z = 1.9
z = 0.95 <<<<<<<<<<<<<<<<<<
Use equations 1 and 2 to check the answers.
1) x - 4y - 2z = 1
3.7 - 4(0.2) - 2(0.95) = 1
3.7 - 0.8 - 1.9 = 1
1 = 1
2) 2x - 5y + 8z = 16
2(3.7) - 5(0.2) + 8(0.95) = 16
7.4 - 1 + 7.6 = 16
16 = 16
Answer: x = 3.7, y = 0.2, z = 0.95