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We are given the equation \( 2A + 3X = 5B \), where:

\[
A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}
\]

We need to find the matrix \( X \).

### Steps:
1. First, isolate \( X \) in the equation:
   \[
   3X = 5B - 2A
   \]
   \[
   X = \frac{5B - 2A}{3}
   \]

2. Compute \( 5B \):
   \[
   5B = 5 \times \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix} = \begin{bmatrix} 10 & -10 \\ 20 & 10 \\ -25 & 5 \end{bmatrix}
   \]

3. Compute \( 2A \):
   \[
   2A = 2 \times \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix} = \begin{bmatrix} 16 & 0 \\ 8 & -4 \\ 6 & 12 \end{bmatrix}
   \]

4. Subtract \( 2A \) from \( 5B \):
   \[
   5B - 2A = \begin{bmatrix} 10 & -10 \\ 20 & 10 \\ -25 & 5 \end{bmatrix} - \begin{bmatrix} 16 & 0 \\ 8 & -4 \\ 6 & 12 \end{bmatrix} = \begin{bmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{bmatrix}
   \]

5. Finally, divide by 3 to get \( X \):
   \[
   X = \frac{1}{3} \times \begin{bmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{bmatrix} = \begin{bmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{bmatrix}
   \]

Thus, the matrix \( X \) is:

\[
X = \begin{bmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{bmatrix}
\]

Do you want more details or have any questions?

### Related Questions:
1. How do you multiply a matrix by a scalar?
2. What is the general method to solve matrix equations of the form \( AX + B = C \)?
3. Can this method be applied to non-square matrices?
4. How would the process change if \( X \) was on both sides of the equation?
5. How does matrix subtraction work element-wise?

### Tip:
Always double-check matrix dimensions before performing operations like multiplication or addition to ensure they are compatible.

If A = [[8, 0], [4, -2], [3, 6]] and B = [[2, -2], [4, 2], [-5, 1]], find the matrix X such that 2A + 3X = 5B.

Learn how to solve the matrix equation 2A + 3X = 5B with a detailed step-by-step breakdown. This problem involves scalar multiplication and subtraction of matrices, key concepts in matrix algebra. Suitable for students in Grades 10-12.

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