The given system of equations can be rewritten in matrix form as:
[-8, -9] [x] = [-16]
[ 6, -9] [y] [-108]
Let's denote the matrix on the left-hand side as A, the column vector on the right-hand side as B, and the column vector containing the variables x and y as X. Then, we have the equation AX = B.
To find X, we need to solve for X in the equation AX = B. We can do this by multiplying both sides by the inverse of matrix A. So, we have X = A^(-1) * B.
First, let's calculate the inverse of matrix A:
A = [-8, -9]
[ 6, -9]
The inverse of a 2x2 matrix can be calculated as:
A^(-1) = (1 / det(A)) * [ -9, 9]
[-6, -8]
where det(A) denotes the determinant of matrix A.
The determinant of matrix A can be found as (-8 * -9) - (6 * -9) = 72 - (-54) = 72 + 54 = 126.
So, the inverse of matrix A is:
A^(-1) = (1 / 126) * [ -9, 9]
[ -6, -8]
Now, let's calculate the product A^(-1) * B:
A^(-1) * B = (1 / 126) * [ -9, 9] * [ -16 ]
[ -6, -8] [ -108]
Multiplying these matrices together, we get:
A^(-1) * B = (1 / 126) * [ (-9 * -16) + (9 * -108) ]
[ (-6 * -16) + (-8 * -108) ]
Simplifying further:
A^(-1) * B = (1 / 126) * [ 144 + (-972) ]
[ 96 + 864 ]
A^(-1) * B = (1 / 126) * [ -828 ]
[ 960 ]
Finally, let's multiply the scalar (1 / 126) by each element in the resulting matrix:
A^(-1) * B = [ -828 / 126 ]
[ 960 / 126 ]
Simplifying further:
A^(-1) * B = [ -6.5714 ]
[ 7.619 ]
So, the solution to the system of equations is x = -6.5714 and y = 7.619.
use a matrix to find the solution of the system of equations
-8-9y=-16
6x-9y=-108
1 answer