Question

How do you solve `x + 9y = 20` and `2x - 4y = 15` using matrices?

Answer 1

`x = 215/22`

`y = 25/22`

Explanation:

Writing the system in matrix form looks like this

`[(1,9),(2,-4)] [(x),(y)] = [(20),(15)]`

Now, the inverse of `[(1,9),(2,-4)]` happens to be `[(2/11,9/22),(1/11,-1/22)]`.

`[(1,9),(2,-4)]^{-1} = 1/(1xx(-4)-9xx2)*[(-4,-2),(-9,1)]^T`

`= [(2/11,9/22),(1/11,-1/22)]`

Multiply that to the left of both sides of the first equation, you will get identity matrix on the left side, and the answer on the right.

`[(2/11,9/22),(1/11,-1/22)][(1,9),(2,-4)] [(x),(y)] = [(2/11,9/22),(1/11,-1/22)][(20),(15)]`

`[(1,0),(0,1)] [(x),(y)] = [(215/22),(25/22)]`

`[(x),(y)] = [(215/22),(25/22)]`

You can check that

`(215/22) + 9(25/22) = 20`

`2(215/22) - 4(25/22) = 15`