The problem asks us to find the value of $x$ given the matrix equation $\begin{bmatrix} 9 & 4 \\ -3 & 5 \end{bmatrix} \begin{bmatrix} 1 \\ -x \end{bmatrix} = \begin{bmatrix} -x \\ -18 \end{bmatrix}$.

1. Problem Description

The problem asks us to find the value of $x$ given the matrix equation

$\begin{bmatrix} 9 & 4 \\ -3 & 5 \end{bmatrix} \begin{bmatrix} 1 \\ -x \end{bmatrix} = \begin{bmatrix} -x \\ -18 \end{bmatrix}$.

2. Solution Steps

First, we multiply the two matrices on the left side:

$\begin{bmatrix} 9 & 4 \\ -3 & 5 \end{bmatrix} \begin{bmatrix} 1 \\ -x \end{bmatrix} = \begin{bmatrix} 9(1) + 4(-x) \\ -3(1) + 5(-x) \end{bmatrix} = \begin{bmatrix} 9 - 4x \\ -3 - 5x \end{bmatrix}$

So, we have the equation:

$\begin{bmatrix} 9 - 4x \\ -3 - 5x \end{bmatrix} = \begin{bmatrix} -x \\ -18 \end{bmatrix}$

This gives us two equations:

$9 - 4x = -x$

$-3 - 5x = -18$

From the first equation, we have:

$9 = 3x$

$x = 3$

From the second equation, we have:

$-5x = -15$

$x = 3$

Since both equations give the same value for $x$, the solution is $x=3$.

3. Final Answer

3