# solve the following simultaneous equations using matrices 3x-6y=24 and -4x+5y=-23

## Assuming you know how to calculate a determinant, the answer is

x = |24 -6| / |3 -6| = -18/-9 = 2
....|-23 5|/ |-4 5|
You can get y by substitution of x=2 in either equation, or from a separate ratio of determinants.
If you are not familiar with the notation, nor why I chose those numbers, nor how to calculate the determinants, then you need to review the subject. Your textbook should contain that information.

## To solve the given simultaneous equations using matrices, we can represent the equations in matrix form as follows:

[ 3 -6 ] [ x ] = [ 24 ]
[ -4 5 ] [ y ] [ -23 ]

Let's denote the coefficient matrix as A, the variable matrix as X, and the constant matrix as B. The equation can then be written in matrix form as AX = B.

Now, in order to find X (the solution matrix), we need to find the inverse of matrix A and multiply it with matrix B.

First, let's find the inverse of matrix A.

The inverse of a 2x2 matrix [ a b ] is given by 1/(ad - bc) * [ d -b ]
[ -c a ]

For our matrix A = [ 3 -6 ]
[ -4 5 ]

The determinant of A (ad - bc) = (3 * 5) - (-6 * -4) = 15 - 24 = -9

Therefore, the inverse of matrix A is:
(1/-9) * [ 5 6 ]
[ 4 3 ]

Now, let's multiply the inverse of A with matrix B:

[ 5 6 ] [ 24 ] = [ x ]
[ 4 3 ] [ -23 ] [ y ]

By multiplying these matrices, we get:
(5 * 24) + (6 * -23) = x
(4 * 24) + (3 * -23) = y

Simplifying, we find:
120 + (-138) = x
96 + (-69) = y

Therefore, x = -18 and y = 27.

Hence, the solution to the simultaneous equations is x = -18 and y = 27.