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).