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