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The given matrix equation is:

\[
3 \begin{bmatrix} 2 & 0 \\ -1 & 5 \end{bmatrix} - 2X = \begin{bmatrix} -10 & 5 \\ 0 & 17 \end{bmatrix}
\]

To solve for \( X \), we can follow these steps:

1. **Distribute the scalar 3** to the elements of the matrix:

\[
3 \begin{bmatrix} 2 & 0 \\ -1 & 5 \end{bmatrix} = \begin{bmatrix} 3 \times 2 & 3 \times 0 \\ 3 \times -1 & 3 \times 5 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ -3 & 15 \end{bmatrix}
\]

2. **Rewrite the equation** after distributing:

\[
\begin{bmatrix} 6 & 0 \\ -3 & 15 \end{bmatrix} - 2X = \begin{bmatrix} -10 & 5 \\ 0 & 17 \end{bmatrix}
\]

3. **Isolate \( 2X \)** by subtracting the matrix on the left-hand side from both sides of the equation:

\[
- 2X = \begin{bmatrix} -10 & 5 \\ 0 & 17 \end{bmatrix} - \begin{bmatrix} 6 & 0 \\ -3 & 15 \end{bmatrix}
\]

4. **Perform the matrix subtraction**:

\[
\begin{bmatrix} -10 & 5 \\ 0 & 17 \end{bmatrix} - \begin{bmatrix} 6 & 0 \\ -3 & 15 \end{bmatrix} = \begin{bmatrix} -10 - 6 & 5 - 0 \\ 0 - (-3) & 17 - 15 \end{bmatrix} = \begin{bmatrix} -16 & 5 \\ 3 & 2 \end{bmatrix}
\]

So, the equation becomes:

\[
-2X = \begin{bmatrix} -16 & 5 \\ 3 & 2 \end{bmatrix}
\]

5. **Solve for \( X \)** by dividing both sides of the equationThe solution for the matrix \( X \) is:

\[
X = \begin{bmatrix} 8 & -2.5 \\ -1.5 & -1 \end{bmatrix}
\]

Would you like more details on any of the steps, or do you have further questions?

Here are 5 related questions for further exploration:

1. How do you solve a matrix equation with variables on both sides?
2. What is the process for multiplying a scalar by a matrix?
3. How do you subtract two matrices of the same dimension?
4. How can you find the inverse of a matrix?
5. What are the properties of matrix addition and scalar multiplication?

**Tip:** When solving matrix equations, ensure that matrix dimensions are compatible for the operations you are performing (e.g., addition, subtraction, multiplication).

Explore how to solve a matrix equation for X using scalar multiplication and subtraction. Understand key concepts in matrix algebra, including matrix operations, properties, and solving linear equations. This solution is ideal for students in Grades 10-12.

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