Question

I have a pre calculus practice problem that I need help with I will add another pic that includes the answer options to this problem.

Answer 1

First, write the matrix equation that represents the given system:

$\begin{bmatrix}-4 & 1 \\ 3 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}9 \\ 7\end{bmatrix}$

If we multiply both sides by the inverse of the coefficient matrix, we get:

$\begin{bmatrix}-4 & 1 \\ 3 & 2\end{bmatrix}^{-1}\begin{bmatrix}-4 & 1 \\ 3 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}-4 & 1 \\ 3 & 2\end{bmatrix}^{-1}\begin{bmatrix}9 \\ 7\end{bmatrix}$

On the left member, the first two matrix factors cancel out. On the right member, find the explicit form of the inverse matrix:

$\begin{bmatrix}x \\ y\end{bmatrix}=-\frac{1}{11}\begin{bmatrix}2 & -1 \\ -3 & -4\end{bmatrix}^{}\begin{bmatrix}9 \\ 7\end{bmatrix}$

Remember that this rule can be used for finding the inverse of a 2x2 matrix:

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

Next, perform the matrix product on the right member of the equation:

$\begin{bmatrix}x \\ y\end{bmatrix}=-\frac{1}{11}^{}\begin{bmatrix}11 \\ -55\end{bmatrix}$

Finally, multiply the matrix on the right member by its coefficient of -1/11:

$\begin{bmatrix}x \\ y\end{bmatrix}=^{}\begin{bmatrix}-1 \\ 5\end{bmatrix}$