We are given a matrix $A$ and its inverse $A^{-1}$. Matrix $A$ is:

$A = \begin{pmatrix} 2 & 3 \\ 5 & 8 \end{pmatrix}$

and its inverse is:

$A^{-1} = \begin{pmatrix} 8 & x \\ -5 & 2 \end{pmatrix}$

We need to find the value of $7x$.

Steps to Solve:

1. Inverse of Matrix Formula: For a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ the inverse is given by: $A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

   For matrix $A = \begin{pmatrix} 2 & 3 \\ 5 & 8 \end{pmatrix}$, we can calculate its inverse using the formula.

2. Calculate the determinant of $A$: The determinant of $A$ is: $\text{det}(A) = (2)(8) - (3)(5) = 16 - 15 = 1$

3. Inverse of $A$: Since the determinant is 1, the inverse of $A$ is: $A^{-1} = \begin{pmatrix} 8 & -3 \\ -5 & 2 \end{pmatrix}$

4. Compare the given inverse: The given inverse is: $A^{-1} = \begin{pmatrix} 8 & x \\ -5 & 2 \end{pmatrix}$

   By comparing the matrices element by element, we get: $x = -3$

5. Calculate $7x$: $7x = 7(-3) = -21$

Thus, the value of $7x$ is $-21$.

Would you like further details on any step?

Here are 5 related questions for deeper understanding:

1. What is the determinant of a 2x2 matrix and how is it used in calculating the inverse?
2. How would the process change if the determinant of the matrix were not 1?
3. Can the inverse of any matrix be found using the same formula?
4. Why is matrix multiplication essential for verifying the correctness of matrix inverses?
5. What are some real-world applications of finding matrix inverses?

Tip: Always check if a matrix has an inverse by verifying if its determinant is non-zero.