Use a matrix to find the solution to the system of equations.
-8x - 8y= -16
6x - 9y = -108
To solve this system of equations using a matrix, we need to set up the coefficient matrix A and the constant matrix B.
The coefficient matrix is:
A = [[-8, -8],
[6, -9]]
The constant matrix is:
B = [[-16],
[-108]]
To find the solution, we can use the formula X = A^-1 * B, where A^-1 is the inverse of matrix A.
The inverse of matrix A can be found by using the formula:
A^-1 = (1 / (ad - bc)) * [[-d, b],
[c, -a]]
Where a, b, c, and d are the elements of matrix A.
In this case, a = -8, b = -8, c = 6, and d = -9.
So, the inverse of matrix A is:
A^-1 = (1 / ((-8 * -9) - (-8 * 6))) * [[-(-9), -(-8)],
[6, -(-8)]]
A^-1 = (1 / (72 + 48)) * [[9, 8],
[6, 8]]
A^-1 = (1 / 120) * [[9, 8],
[6, 8]]
A^-1 = [[9/120, 8/120],
[6/120, 8/120]]
A^-1 = [[3/40, 1/15],
[1/20, 1/15]]
Now, we can calculate the solution matrix X by multiplying A^-1 and B:
X = [[3/40, 1/15],
[1/20, 1/15]] * [[-16],
[-108]]
X = [[(3/40 * -16) + (1/15 * -108)],
[(1/20 * -16) + (1/15 * -108)]]
X = [[(-48/40) + (-108/15)],
[(-16/20) + (-108/15)]]
X = [[-60/40 + -72/5],
[-8/20 + -72/5]]
X = [[(-150 + -144)/40],
[(-4 + -144)/20]]
X = [[-294/40],
[-148/20]]
X = [[-7.35],
[-7.4]]
Therefore, the solution to the system of equations is x = -7.35 and y = -7.4.