Solve this system of equations.  {2x-2y+z=3 {5y-z=-31 {x+3y+2z=-21 (All in one bracket)

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Solve the system `{[2x-2y+z=3],[5y-z=-31],[x+3y+2z=-21]}`

We can use linear combinations to solve. We choose to eliminate z:

Add the first equation to the second equation: 2x+3y=-28

Add the third equation to twice the second equation: x+13y=-83

Solve the new system:

2x+3y=-28x+13y=-83    Subtract twice the second equation from the first:

-23y=138...

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Solve the system `{[2x-2y+z=3],[5y-z=-31],[x+3y+2z=-21]}`

We can use linear combinations to solve. We choose to eliminate z:

Add the first equation to the second equation: 2x+3y=-28

Add the third equation to twice the second equation: x+13y=-83

Solve the new system:

2x+3y=-28
x+13y=-83    Subtract twice the second equation from the first:

-23y=138 ==>y=-6

y=-6==>2x-18=-28 ==> 2x=-10 ==>x=-5

x=-5,y=-6==>-30-z=-31==>-z=-1==>z=1

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The solution is x=-5,y=-6,z=1

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