Question

How do you solve `9x-5y=-44` and `4x-3y=-18` using matrices?

Answer 1

The answer (in matrix form) is: `((1,0, -6),(0,1, 2))`.

Explanation:

We can translate the given equations into matrix notation by transcribing the coefficients to elements of a 2x3 matrix:

`((9, -5, -44), (4, -3, -18))`

Divide the second row by 4 to get a one in the "x column."

`((9, -5, -44), (1, -3/4, -9/2))`

Add -9 times the second row to the top row to get a zero in the "x column." We'll also revert the second row back to its previous form by multiplying by 4 again.

`((0, 7/4, -7/2), (4, -3, -18))`

Multiply the top row by `4/7` to get a 1 in the "y column."

`((0, 1, -2), (4, -3, -18))`

We now have an answer for y. In order to solve for x, we add 3 times the first row to the second row.

`((0, 1, -2), (4, 0, -24))`

Then divide the second row by 4.

`((0, 1, -2), (1, 0, -6))`

And we finish by reversing the rows since it's traditional to show your final solution in the form of an identity matrix and an auxiliary column.

`((1, 0, -6), (0, 1, -2))`

This is equivalent to the set of equations:
`x = -6`
`y = -2`