Use the matrix tool to solve the system of equations. Enter the answer as an ordered pair. 8x + 5y = 9, -x + y = 7.

Steps to Solve

1. Set up the equations in matrix form

The given system of equations can be expressed in matrix form. The equations are:

$$ \begin{align*}

1. & \quad 8x + 5y = 9 \
2. & \quad -x + y = 7 \end{align*} $$

This corresponds to the matrix equation ( AX = B ):

$$ A = \begin{pmatrix} 8 & 5 \ -1 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 9 \ 7 \end{pmatrix} $$

2. Calculate the inverse of matrix A

To solve for ( X ), we need to find the inverse of matrix ( A ) (if it exists). The formula for the inverse of a 2x2 matrix

$$ \begin{pmatrix} a & b \ c & d \end{pmatrix} $$

is given by

$$ \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} $$

Here, ( a = 8 ), ( b = 5 ), ( c = -1 ), and ( d = 1 ).

3. Find the determinant of A

Calculate the determinant:

$$ \text{det}(A) = (8)(1) - (5)(-1) = 8 + 5 = 13 $$

Since the determinant is not zero, ( A ) is invertible.

4. Find the inverse of A

Using the inverted matrix formula:

$$ A^{-1} = \frac{1}{13} \begin{pmatrix} 1 & -5 \ 1 & 8 \end{pmatrix} $$

5. Multiply the inverse of A by B

Now, calculate ( X = A^{-1}B ):

$$ X = \frac{1}{13} \begin{pmatrix} 1 & -5 \ 1 & 8 \end{pmatrix} \begin{pmatrix} 9 \ 7 \end{pmatrix} $$

Calculating the multiplication:

$$ X = \frac{1}{13} \begin{pmatrix} (1)(9) + (-5)(7) \ (1)(9) + (8)(7) \end{pmatrix} = \frac{1}{13} \begin{pmatrix} 9 - 35 \ 9 + 56 \end{pmatrix} = \frac{1}{13} \begin{pmatrix} -26 \ 65 \end{pmatrix} $$

6. Simplify to find x and y

Now simplify:

$$ X = \begin{pmatrix} -2 \ 5 \end{pmatrix} $$

Thus, ( x = -2 ) and ( y = 5 ).