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The given problem asks us to find matrix \( X \) such that the equation \( 2A + 3X = 5B \) holds, where:

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

### Step 1: Rearrange the equation
We start with the equation:
\[
2A + 3X = 5B
\]
Isolate \( X \) by subtracting \( 2A \) from both sides:
\[
3X = 5B - 2A
\]
Now, divide by 3:
\[
X = \frac{5B - 2A}{3}
\]

### Step 2: Compute \( 5B \) and \( 2A \)
First, calculate \( 5B \):
\[
5B = 5 \times \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}
= \begin{bmatrix} 10 & -10 \\ 20 & 10 \\ -25 & 5 \end{bmatrix}
\]

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

### Step 3: Subtract \( 2A \) from \( 5B \)
Now subtract \( 2A \) from \( 5B \):
\[
5B - 2A = \begin{bmatrix} 10 & -10 \\ 20 & 10 \\ -25 & 5 \end{bmatrix}
- \begin{bmatrix} 16 & 0 \\ 8 & -4 \\ 6 & 12 \end{bmatrix}
= \begin{bmatrix} 10 - 16 & -10 - 0 \\ 20 - 8 & 10 - (-4) \\ -25 - 6 & 5 - 12 \end{bmatrix}
= \begin{bmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{bmatrix}
\]

### Step 4: Divide by 3 to find \( X \)
Finally, divide each element by 3:
\[
X = \frac{1}{3} \times \begin{bmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{bmatrix}
= \begin{bmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{bmatrix}
\]

Thus, the matrix \( X \) is:
\[
X = \begin{bmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{bmatrix}
\]

Let me know if you need further details or explanations!

Here are five related questions:
1. How do you multiply a matrix by a scalar?
2. What is the method to subtract two matrices?
3. How is matrix division by a scalar done?
4. How do you find the inverse of a matrix, if it exists?
5. What are the properties of matrix multiplication?

**Tip:** Always ensure the matrices have the same dimensions before adding or subtracting them!

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 step by step. This problem covers key concepts such as scalar multiplication and matrix subtraction, suitable for high school students in Grades 10-12.

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