In this problem, you are given three matrices $A$, $B$, and $C$. The equation provided is:

$A + B = C$

Where:

$A = \begin{pmatrix} 2 & x \\ 4 & y \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 3 \\ -6 & 2x \end{pmatrix}, \quad C = \begin{pmatrix} 3 & 5 \\ -2 & 5 \end{pmatrix}$

We need to find the values of $x$ and $y$.

Step-by-Step Solution:

We begin by adding the matrices $A$ and $B$ element-wise and equating them to the corresponding elements in matrix $C$:

$A + B = \begin{pmatrix} 2 & x \\ 4 & y \end{pmatrix} + \begin{pmatrix} 1 & 3 \\ -6 & 2x \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ -2 & 5 \end{pmatrix}$

This gives us the following system of equations for each element:

1. First row, first column: $2 + 1 = 3 \quad \text{(This is correct, so no new information from here.)}$

2. First row, second column: $x + 3 = 5 \quad \Rightarrow \quad x = 2$

3. Second row, first column: $4 + (-6) = -2 \quad \text{(This is correct, so no new information from here.)}$

4. Second row, second column: $y + 2x = 5$ Substitute $x = 2$ into this equation: $y + 2(2) = 5 \quad \Rightarrow \quad y + 4 = 5 \quad \Rightarrow \quad y = 1$

Final Answer:

The values of $x$ and $y$ are: $x = 2, \quad y = 1$

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

Here are 5 related questions you might find interesting:

1. How do you add two matrices in general?
2. How do you solve a system of linear equations?
3. What are the properties of matrix addition?
4. Can matrix multiplication be applied in this context?
5. How would the result change if the matrices were larger or of a different dimension?

Tip: Always make sure the matrices are of the same dimension when adding or subtracting them.