x+y+z=6; 2x-y+5z=15; 3x-y-2z=-5
x+y+z= 6
2x-y+5z= 15
3x-y-2z= -5
1) x + y + z = 6
2) 2x - y + 5z = 15
3) 3x - y - 2z = -5
Multiply equation 1 by 5.
5(x + y + z) = 6 * 5
4) 5x + 5y + 5z = 30
Subtract equation 2 from equation 4.
5x + 5y + 5z = 30
-(2x - y + 5z = 15)
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3x + 6y = 15
5) 3x + 6y = 15
Multiply equation 2 by 2.
2(2x - y + 5z) = 15 * 2
6) 4x - 2y + 10z = 30
Multiply equation 3 by 5.
5(3x - y - 2z) = -5 * 5
7) 15x - 5y - 10z = -25
Add equation 7 to equation 6.
4x - 2y + 10z = 30
+(15x - 5y - 10z = -25)
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19x - 7y = 5
8) 19x - 7y = 5
Multiply equation 5 by 7.
7(3x + 6y) = 15 * 7
9) 21x + 42y = 105
Multiply equation 8 by 6.
6(19x - 7y) = 5 * 6
10) 114x - 42y = 30
Add equation 10 to equation 9.
21x + 42y = 105
+(114x - 42y = 30)
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135x = 135
135x = 135
x = 135/135
x = 1 <<<<<<<<<<<<<<<<<<<
Use equation 5 to solve for y.
3x + 6y = 15
3(1) + 6y = 15
3 + 6y = 15
6y = 15 - 3
6y = 12
y = 12/6
y = 2 <<<<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
3x - y - 2z = -5
3(1) - 2 - 2z = -5
3 - 2 - 2z = -5
1 - 2z = -5
-2z = -5 - 1
-2z = -6
z = -6/-2
z = 3 <<<<<<<<<<<<<<<<<<<
Use all three of the given equations to check the answers.
1) x + y + z = 6
1 + 2 + 3 = 6
6 = 6
2) 2x - y + 5z = 15
2(1) - 2 + 5(3) = 15
2 - 2 + 15 = 15
15 = 15
3) 3x - y - 2z = -5
3(1) - 2 - 2(3) = -5
3 - 2 - 6 = -5
-5 = -5
The answer: x = 1, y = 2, z = 3