solve 5x=3Y=13 and 4x+7y=-8 using matrices

Let A be a 2x2 matrix such that A = |a b|

|c d|

Then the inverse of A is given by,

A^(-1) = (1/det(A)) * |d -b|

|-c a|

Our equations are:

5x + 3y = 13 | 5 3|| x | = |13|

4x + 7y = -8 | 4 7|| y | = | -8|

Which in matrix form is: AX = R,

Where X is the unknowns matrix, [x y], and R is the constants matrix [13 -8].

Our determinant is A = |5 3|

|4 7|

det(A) = ad – bc = 5*7 – 3*4 = 35 – 12 = 23

__det(A) = 23__

using the definition for A^(-1) above,

A^(-1) = (1/23)*|7 -3|

|-4 5|

And,

X = A^(-1) * R = (1/23)*|7 -3| * |13| = (1/23)*|91 + 24| = (1/23)*|115| = | 5|

|-4 5| | -8| |-52 -40| |-92| |-4|

X = | 5|

|-4|

__Solution: x = 5, y = -4__