The problem asks to find a 2x2 matrix $X$ given the equation $3B + X = 2C$, where matrices $B$ and $C$ are provided as:

$B = \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 2 & -5 \\ 1 & 7 \end{bmatrix}$

Steps to solve:

1. Rearrange the equation $3B + X = 2C$ to solve for $X$: $X = 2C - 3B$

2. Compute $3B$: $3B = 3 \times \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix} = \begin{bmatrix} 3 & -12 \\ 9 & 15 \end{bmatrix}$

3. Compute $2C$: $2C = 2 \times \begin{bmatrix} 2 & -5 \\ 1 & 7 \end{bmatrix} = \begin{bmatrix} 4 & -10 \\ 2 & 14 \end{bmatrix}$

4. Subtract $3B$ from $2C$: $X = \begin{bmatrix} 4 & -10 \\ 2 & 14 \end{bmatrix} - \begin{bmatrix} 3 & -12 \\ 9 & 15 \end{bmatrix} = \begin{bmatrix} 4 - 3 & -10 + 12 \\ 2 - 9 & 14 - 15 \end{bmatrix}$ $X = \begin{bmatrix} 1 & 2 \\ -7 & -1 \end{bmatrix}$

Thus, the matrix $X$ is:

$X = \begin{bmatrix} 1 & 2 \\ -7 & -1 \end{bmatrix}$

Would you like further clarification on any steps?

Here are 5 related questions you might consider:

1. How do you calculate the determinant of matrix $X$?
2. What is the inverse of matrix $B$, if it exists?
3. How do matrix operations (like addition and scalar multiplication) affect determinants?
4. Can you find the eigenvalues of matrix $B$?
5. What are the conditions for two matrices to be invertible?

Tip: When solving matrix equations, always carefully apply operations to each element to avoid errors.