Question

How do you solve using gaussian elimination or gauss-jordan elimination, `3x + y =1`, `-7x - 2y = -1`?

Answer 1

This can be rewritten into an augmented matrix form:

`[(3,1,|,1),(-7,-2,|,-1)]`

where each entry is a coefficient associated with each variable, and the last column contains the answers to each equality.

To get this into row echelon form, we need to get leading `1`'s in each row such that the matrix is shaped like an upside-down stepladder and entries below each `1` are `0`. If any rows are all `0`'s, then it has to be at the bottom.

To get it into reduced-row echelon form, get all numbers above AND below each leading `1` to be `0`.

I am using the notation where the rightmost indicated row is the one that is operated upon.

`stackrel(3R_1 + R_2" ")(->)[(3,1,|,1),(2,1,|,2)]`

`stackrel(-R_2 + R_1" ")(->)[(1,0,|,-1),(2,1,|,2)]`

`stackrel(-2R_1 + R_2" ")(->)[(color(blue)(1),0,|,color(blue)(-1)),(0,color(blue)(1),|,color(blue)(4))]`

From here we can just bring it back to algebraic form to get:

`color(blue)(x = -1)`
`color(blue)(y = 4)`

We can check to see that this is correct by plugging these back in:

`3(-1) + (4) = -3 + 4 = 1`
`-7(-1) - 2(4) = 7 - 8 = -1`