I am labeling the equations 1,2,3
1. x - y + 3z = 13
2. 4x + y + 2z = 17
3. 3x + 2y + 2z = 1
(2-3)
4x + y + 2z = 17
-3x - 2y -2z = -1
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x - y = 16
x = 16 - y
++++++++++++++++++++++++
substitute x into 1.
x - y + 3z = 13
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16 - y -y + 3z = 13
-2y + 3z = -3 label (1a)
++++++++++++++++++++++++++++++
substitute x into 2.
4x + y +2z = 17
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4(16 - y) + y + 2z = 17
64 - 4y + y + 2z = 17
-3y + 2z = -47 label this (2a.)
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substitute x into 3
3x + 2y +2z = 1
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3(16 - y) + 2y + 2z = 1
48 -3y + 2y + 2z = 1
-y + 2z = -47 label (3a.)
++++++++++++++++++++++++++++++++
(2a - 3a) lets us solve for y
-3y + 2z = -47
+ y - 2z = 47
-2y = 0
y = 0
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substitute y = 0 into 1a. (because we used 2a and 3a to solve for y
-2y + 3z = -3
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-2(0) + 3z = -3
z = -1
++++++++++++++++++++++++++++++++++++++
substitute y = 0 and z = -1 into 1.
x - y +3z = 13
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x - 0 + 3 (-1) = 13
x - 3 = 13
x = 10
+++++++++++++++++++++++++++++++++==
x = 10, y = 0 , z = -1