The problem asks us to find the value of $x$ given the matrix equation: $$ \begin{bmatrix} 6 & -2 \\ -7 & 4 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \frac{1}{3x} \begin{bmatrix} 36 \\ -12 \end{bmatrix} $$

1. Problem Description

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

$$\begin{bmatrix} 6 & -2 \\ -7 & 4 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \frac{1}{3x} \begin{bmatrix} 36 \\ -12 \end{bmatrix}$$

2. Solution Steps

First, we perform the matrix multiplication on the left side:

$$\begin{bmatrix} 6 & -2 \\ -7 & 4 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} (6)(2) + (-2)(3) \\ (-7)(2) + (4)(3) \end{bmatrix} = \begin{bmatrix} 12 - 6 \\ -14 + 12 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}$$

So the equation becomes:

$$\begin{bmatrix} 6 \\ -2 \end{bmatrix} = \frac{1}{3x} \begin{bmatrix} 36 \\ -12 \end{bmatrix}$$

This means:

$$6 = \frac{36}{3x}$$

$$-2 = \frac{-12}{3x}$$

We can use either equation to solve for $x$. Let's use the first equation:

$$6 = \frac{36}{3x}$$

Multiply both sides by $3x$:

$$18x = 36$$

Divide both sides by 18:

$$x = \frac{36}{18} = 2$$

Now, let's check if the second equation gives the same result:

$$-2 = \frac{-12}{3x}$$

Multiply both sides by $3x$:

$$-6x = -12$$

Divide both sides by -6:

$$x = \frac{-12}{-6} = 2$$

Both equations give the same result, $x=2$.

3. Final Answer

The value of $x$ is 2.