To solve for the value of [tex]\( x \)[/tex] in the matrix equation
[tex]\[
\left[\begin{array}{cc} 2x & 5 \\ 8 & x \end{array}\right] = \left[\begin{array}{cc} 6 & -2 \\ 7 & 3 \end{array}\right],
\][/tex]
we need to compare each corresponding element of the two matrices to set up and solve a series of equations.
### Step 1: Compare the elements.
From the matrices given, we get:
1. [tex]\( 2x = 6 \)[/tex] from the (1,1) position.
2. [tex]\( 5 = -2 \)[/tex] from the (1,2) position.
3. [tex]\( 8 = 7 \)[/tex] from the (2,1) position.
4. [tex]\( x = 3 \)[/tex] from the (2,2) position.
### Step 2: Solve for [tex]\( x \)[/tex]
1. From the equation [tex]\( 2x = 6 \)[/tex]:
[tex]\[
2x = 6 \implies x = \frac{6}{2} \implies x = 3
\][/tex]
2. The equations [tex]\( 5 = -2 \)[/tex] and [tex]\( 8 = 7 \)[/tex] are inconsistent and indicate no meaningful relationship for the problem.
3. From the equation [tex]\( x = 3 \)[/tex]:
[tex]\[
x = 3
\][/tex]
### Conclusion:
Both equations [tex]\( 2x = 6 \)[/tex] and [tex]\( x = 3 \)[/tex] yield the same value of [tex]\( x \)[/tex]. Therefore, the solution to the given matrix equation is
[tex]\[
x = 3
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{3}\)[/tex].
