Question

How do you solve the system `2x - 3y = -11` and `3x + 2y = 29`?

Answer 1

Let `M = ((2,3),(-3,2))`. Then `det(M) = 2^2+3^2=13`

`1/13((2,3),(-3,2))((-11),(29)) = 1/13((65),(91)) = ((5),(7))`

`x=5` and `y=7`

Explanation:

Since the second equation swaps the coefficients of `x` and `y` with exactly one sign change, we can tell that the second equation is of a line perpendicular to that described by the first.

Hence all we need to do to get `x` and `y` is to multiply `((-11),(29))` by a rotation matrix constructed from the coefficients of `x` and `y`.

graph{(2x-3y+11)(3x+2y-29) = 0 [-16.18, 23.82, -4.16, 15.84]}