Question

How do you solve using gaussian elimination or gauss-jordan elimination, `x-y+3z=13`, `4x+y+2z=17`, `3x+2y+2z=1`?

Answer 1

`x=7`
`y=-9`
`z=-1`

Explanation:

Initial Augmented Matrix:
#[ ( 1.00, -1.00, 3.00, 13.00), ( 4.00, 1.00, 2.00, 17.00), ( 3.00, 2.00, 2.00, 1.00) ]#

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 1.00 so pivot entry = 1
#[ ( 1.00, -1.00, 3.00, 13.00), ( 4.00, 1.00, 2.00, 17.00), ( 3.00, 2.00, 2.00, 1.00) ]#

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

#[ ( 1.00, -1.00, 3.00, 13.00), ( 0.00, 5.00, -10.00, -35.00), ( 0.00, 5.00, -7.00, -38.00) ]#

Pivot `color(black)(2)`
Pivot Row 2 reduced by dividing all entries by 5.00 so pivot entry = 1
#[ ( 1.00, -1.00, 3.00, 13.00), ( 0.00, 1.00, -2.00, -7.00), ( 0.00, 5.00, -7.00, -38.00) ]#

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

#[ ( 1.00, 0.00, 1.00, 6.00), ( 0.00, 1.00, -2.00, -7.00), ( 0.00, 0.00, 3.00, -3.00) ]#

Pivot `color(black)(3)`
Pivot Row 3 reduced by dividing all entries by 3.00 so pivot entry = 1
#[ ( 1.00, 0.00, 1.00, 6.00), ( 0.00, 1.00, -2.00, -7.00), ( 0.00, 0.00, 1.00, -1.00) ]#

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, 7.00), ( 0.00, 1.00, 0.00, -9.00), ( 0.00, 0.00, 1.00, -1.00) ]#