Question

How do you solve using gaussian elimination or gauss-jordan elimination, `4x - y + 3z = 12`, `x + 4y + 6z = -32`, `5x + 3y + 9z = 20`?

Answer 1

There is no solution since the three equations are not independent.

Explanation:

Conversion of Given Equations into Augmented Matrix:
`[ ( 4.00, -1.00, 3.00, 12.00), ( 1.00, 4.00, 6.00, -32.00), ( 5.00, 3.00, 9.00, 20.00) ]`

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Pivot Action `n`
pivot row = n; pivot column = n; pivot entry augmented matrix entry at (n,n)

1. convert pivot n row so pivot entry = 1
2. adjust non-pivot rows so entries in pivot column = 0

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Pivot `color(black)(1)`
Pivot Row 1 reduced by dividing all entries by 4.00 so pivot entry = 1
`[ ( 1.00, -0.25, 0.75, 3.00), ( 1.00, 4.00, 6.00, -32.00), ( 5.00, 3.00, 9.00, 20.00) ]`

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 1 from each non-pivot row

`[ ( 1.00, -0.25, 0.75, 3.00), ( 0.00, 4.25, 5.25, -35.00), ( 0.00, 4.25, 5.25, 5.00) ]`

Pivot `color(black)(2)`
Pivot Row 2 reduced by dividing all entries by 4.25 so pivot entry = 1
`[ ( 1.00, -0.25, 0.75, 3.00), ( 0.00, 1.00, 1.24, -8.24), ( 0.00, 4.25, 5.25, 5.00) ]`

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 2 from each non-pivot row

`[ ( 1.00, 0.00, 1.06, 0.94), ( 0.00, 1.00, 1.24, -8.24), ( 0.00, 0.00, 0.00, 40.00) ]`

Pivot `color(black)(3)`
Pivot row 3 has a 0-valued pivot entry; exchange attempted
No subsequent rows have a non-zero value in the pivot column

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Given equations allow for no solution

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