Question

How do you solve using gaussian elimination or gauss-jordan elimination, `3x+2y = -9`, `-10x + 5y = - 5`?

Answer 1

See below

Explanation:

First create the augmented matrix:

`((3,2),(-10,5))((-9),(-5))`

For Gaussian elimination , we eliminate by combining rows to get a lower triangle matrix.

`R2 to R2 + 10/3 R1`
`((3,2),(0,35/3))((-9),(-35))`

Then back substitute, so from the second row:

`35/3 y = - 35 implies y -3`

From first row:

`3x + 2y = - 9 implies x = (-9 - 2(-3))/(3) = -1`

For Gauss-Jordan elimination , we row reduce to the identity matrix ie `A mathbf x = mathbf b to I mathbf x = mathbf b'`. So the first step can be the same:

`R2 to R2 + 10/3 R1`
`((3,2),(0,35/3))((-9),(-35))`

Then we can do this:
`R1 to R1 - R2 cdot 6/35`
`((3,0),(0,35/3))((-3),(-35))`

Finally we divide each row by its pivot, ie:

`((3/3,0),(0,(35/3)/(35/3)))((-3/3),(-35/(35/3)))`

`implies ((1,0),(0,1))((-1),(-3))`