Use a matrix to find the solution to the system of equations.

{-8x - 8y = -16
{6x - 9y = -108
A.(-6,8)
B.(6,8)
C.(8,-6)
D.(6,-8)

I just used this answer and it is correct

First I'd factor out what I could to make the system

x+y = 2
2x-3y = -36
So, now you have
AX=B
where
A = {{1,1},{2,-3}} and B={{2},{-36}}
so A^-1 = 1/5 {{3,1},{2,-1}}
and we have
X = A^-1*B = 1/5 {{3,1},{2,-1}}*{{2},{-36}} = {{-6},{8}}
so (-6,8) is the solution

To use a matrix to find the solution to the given system of equations, we can set up an augmented matrix.

The augmented matrix for this system is:

[-8 -8 | -16]
[ 6 -9 | -108]

To solve this system using matrices, we can use row operations to simplify the matrix. The goal is to get the matrix into row-echelon form or reduced row-echelon form.

Let's start by simplifying the first row by multiplying it by 6 and adding it to the second row:

[ -8 -8 | -16]
[ 0 -57 | -224]

We can now simplify the second row by dividing it by -57 to get:

[ -8 -8 | -16]
[ 0 1 | 224/57]

Now, let's simplify the first row by adding 8 times the second row:

[ -8 0 | -16 + 8(224/57)]
[ 0 1 | 224/57]

This simplifies to:

[ -8 0 | -64/57 ]
[ 0 1 | 224/57 ]

Finally, let's simplify the first row by dividing it by -8 to get:

[ 1 0 | 8/57 ]
[ 0 1 | 224/57 ]

This matrix is now in row-echelon form.

The solution to the system of equations is x = 8/57 and y = 224/57.

Therefore, the correct answer is A. (8/57, 224/57).

To use a matrix to solve the system of equations, we can represent the system of equations in matrix form as:

[A] [x] = [b]

Where:
[A] is the coefficient matrix,
[x] is the variables matrix,
[b] is the constant matrix.

For the given system of equations:
{-8x - 8y = -16
{6x - 9y = -108

We can rewrite it in matrix form as:

[[-8, -8], [6, -9]] [x, y] = [-16, -108]

To solve this matrix equation, we can use the inverse of the coefficient matrix A.

Let's find the inverse of the matrix A:

[A] = [[-8, -8], [6, -9]]

The inverse of the matrix A, denoted as [A]^-1, can be calculated as:

[A]^-1 = (1 / (ad - bc)) * [[-d, b], [-c, a]]

Where a = -8, b = -8, c = 6, and d = -9.

Let's calculate the inverse of the matrix A:

[A]^-1 = (1 / (-8*(-9) - (-8)*6)) * [[-(-9), -8], [-6, -8]]
= (1 / (72 + 48)) * [[9, -8], [-6, -8]]
= (1 / 120) * [[9, -8], [-6, -8]]
= [[9/120, -8/120], [-6/120, -8/120]]
= [[3/40, -1/15], [-1/20, -1/15]]

Now, we can calculate the variables matrix [x, y] by multiplying the inverse of A with the constant matrix [b]:

[x, y] = [A]^-1 * [b]

[x, y] = [[3/40, -1/15], [-1/20, -1/15]] * [-16, -108]

To perform this matrix multiplication, let's calculate each entry:

x = (3/40) * (-16) + (-1/15) * (-108)
= (-48/40) + (8/15)
= -6 + (32/40) + (8/15)
= -6 + (4/5) + (8/15)
= -6 + (60/75) + (40/75)
= -6 + (100/75)
= -6 + (4/3)
= -6 + 1 1/3
= -6 + 1 + 1/3
= -5 + 1/3

y = (-1/20) * (-16) + (-1/15) * (-108)
= (16/20) + (108/15)
= (4/5) + (108/15)
= (12/15) + (108/15)
= (120/15) + (108/15)
= (120 + 108) / 15
= 228 / 15
= 76 / 5
= 15 1/5

Therefore, the solution to the system of equations is [x, y] = [-5 1/3, 15 1/5].

Comparing this solution with the given answer choices, we can see that none of the choices match.