Question

How do you solve the system `4x-y=1, x+2y=7` using matrix equation?

Answer 1

`((x),(y)) = ((1),(3))`

Explanation:

In matrix form it's this:

`((4,-1),(1,2))((x),(y)) = ((1),(7))`

You can the go through the formality of getting the inverse of the matrix (assuming of course it has one). If we start with the determinant:

`det ((4,-1),(1,2)) = 4(2) - 1(-1) = 9` so the matrix is invertible!

And we know that the inverse is:

`((4,-1),(1,2))^(-1) = 1/9 ((2,1),(-1,4))`

And we can say that:

`color(red)( 1/9 ((2,1),(-1,4))) color(green)( ((4,-1),(1,2))) ((x),(y)) = color(red)( 1/9 ((2,1),(-1,4))) ((1),(7))`

And because the red and the green terms combine as the identity matrix:

`implies ((x),(y)) = 1/9 ((2,1),(-1,4)) ((1),(7))`

`implies ((x),(y)) = 1/9 ((9),(27)) = ((1),(3))`

Generally speaking solving systems is easier using Row Reduction but in the case of a 2x2 it is also pretty easy to work through this method.