Question

How do you solve using gaussian elimination or gauss-jordan elimination, `4x_1 + 5x_2 + 2x_3 = 11`, `2x_2 + 3x_3 - 4x_4 = -2`, `2x_1 + x_2 + 3x_4 = 12`, `x_1 + x_3 + x_4 = 9`?

Answer 1

`x_1=2`
`x_2=-1`
`x_3=4`
`x_4=3`

Explanation:

Initial Augmented Matrix:
`[ ( 4.00, 5.00, 2.00, 0.00, 11.00), ( 0.00, 2.00, 3.00, -4.00, -2.00), ( 2.00, 1.00, 0.00, 3.00, 12.00), ( 1.00, 0.00, 1.00, 1.00, 9.00) ]`

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

Pivot `color(black)(1)`
Pivot Row 1 reduced by dividing all entries by 4.00 so pivot entry = 1
`[ ( 1.00, 1.25, 0.50, 0.00, 2.75), ( 0.00, 2.00, 3.00, -4.00, -2.00), ( 2.00, 1.00, 0.00, 3.00, 12.00), ( 1.00, 0.00, 1.00, 1.00, 9.00) ]`

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

`[ ( 1.00, 1.25, 0.50, 0.00, 2.75), ( 0.00, 2.00, 3.00, -4.00, -2.00), ( 0.00, -1.50, -1.00, 3.00, 6.50), ( 0.00, -1.25, 0.50, 1.00, 6.25) ]`

Pivot `color(black)(2)`
Pivot Row 2 reduced by dividing all entries by 2.00 so pivot entry = 1
`[ ( 1.00, 1.25, 0.50, 0.00, 2.75), ( 0.00, 1.00, 1.50, -2.00, -1.00), ( 0.00, -1.50, -1.00, 3.00, 6.50), ( 0.00, -1.25, 0.50, 1.00, 6.25) ]`

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

`[ ( 1.00, 0.00, -1.38, 2.50, 4.00), ( 0.00, 1.00, 1.50, -2.00, -1.00), ( 0.00, 0.00, 1.25, 0.00, 5.00), ( 0.00, 0.00, 2.37, -1.50, 5.00) ]`

Pivot `color(black)(3)`
Pivot Row 3 reduced by dividing all entries by 1.25 so pivot entry = 1
`[ ( 1.00, 0.00, -1.38, 2.50, 4.00), ( 0.00, 1.00, 1.50, -2.00, -1.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 2.37, -1.50, 5.00) ]`

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 3 from each non-pivot row

`[ ( 1.00, 0.00, 0.00, 2.50, 9.50), ( 0.00, 1.00, 0.00, -2.00, -7.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 0.00, -1.50, -4.50) ]`

Pivot `color(black)(4)`
Pivot Row 4 reduced by dividing all entries by -1.50 so pivot entry = 1
`[ ( 1.00, 0.00, 0.00, 2.50, 9.50), ( 0.00, 1.00, 0.00, -2.00, -7.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 0.00, 1.00, 3.00) ]`

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 4 from each non-pivot row

`[ ( 1.00, 0.00, 0.00, 0.00, 2.00), ( 0.00, 1.00, 0.00, 0.00, -1.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 0.00, 1.00, 3.00) ]`