Find z in the unique solution of the systemx + 2y + 3z =1-x - y + 3z = 2-6x + y + z = -2
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We have three equations to solve for z.
x + 2y + 3z =1 ...(1)
-x - y + 3z = 2 ...(2)
-6x + y + z = -2 ...(3)
x + 2y + 3z =1
=> x = 1 - 2y - 3z ...(4)
substitute in (2)
-x - y + 3z = 2
=> -1 + 2y + 3z - y + 3z = 2
=> y + 6z = 3
=> y = 3 - 6z
Substitute in (4)
=> x = 1 - 6 + 12z - 3z
=> x = -5 + 9z ... (5)
Substitute x and y from (5) and (4) resp. in (3)
-6(-5 + 9z) + 3 - 6z + z = -2
=> 30 - 54z + 3 - 6z + z = -2
=> -59z = -35
=> z = 35/59
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We'll add 1st and the 2nd equations to eliminate x:
x + 2y + 3z - x - y + 3z = 1 + 2
We'll combine like terms:
y + 6z = 3 (4)
We'll multiply the 1st equation by 6:
6x + 12y + 18z = 6 (5)
We'll add (5) and (3) to eliminate x:
6x + 12y + 18z - 6x + y + z = 6 - 2
13y + 19z = 4(6)
We'll eliminate y from (4) and (6)
We'll multiply (4) by -13 and (6) by 6:
13y - 78z = -39 (7)
We'll add (7) and (6):
-13y - 78z + 13y + 19z= -39+4
We'll eliminate and combine like terms:
-59z = -35
z = 35/59
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