https://file.aiquickdraw.com/mathbot/file/1728191846432nd31r0zs.jpg

The given image shows a matrix equation where you need to determine the values of \(x\) and \(y\). Let's go step by step to solve this.

### The Matrix Equation:
\[
\begin{bmatrix} 2x & -1 & 4 \\ 5 & 0 & -3 \end{bmatrix} + \begin{bmatrix} -2 & y & -1 \\ -1 & 2 & 1 \end{bmatrix} = \begin{bmatrix} 2 & -2 & 5 \\ 4 & 2 & -2 \end{bmatrix}
\]

To solve this, we simply add the corresponding elements of the matrices on the left-hand side to match the elements on the right-hand side.

### Step 1: Add Corresponding Elements
The given equation can be broken down element-wise as:

1. First row, first column:
   \[
   2x + (-2) = 2
   \]
   Solving for \(x\):
   \[
   2x - 2 = 2 \implies 2x = 4 \implies x = 2
   \]

2. First row, second column:
   \[
   -1 + y = -2
   \]
   Solving for \(y\):
   \[
   y = -2 + 1 = -3
   \]

### Step 2: Check Consistency
Now that we have values for \(x\) and \(y\), let's ensure that the entire equation balances.

For the rest of the elements:
- First row, third column:
  \[
  4 + (-1) = 5
  \]
  \[
  4 - 1 = 5
  \]
  This is consistent.

- Second row, first column:
  \[
  5 + (-1) = 4
  \]
  \[
  4 = 4
  \]
  This is consistent.

- Second row, second column:
  \[
  0 + 2 = 2
  \]
  This is consistent.

- Second row, third column:
  \[
  -3 + 1 = -2
  \]
  This is consistent.

### Final Answer:
\[
x = 2, \quad y = -3
\]

Would you like more details or have any questions about this solution?

### Related Questions:
1. How do we add matrices with different dimensions?
2. What properties of matrix addition make it commutative?
3. How do we solve a system of linear equations using matrices?
4. What are the properties of matrix scalar multiplication?
5. How does matrix addition differ from matrix multiplication?

### Tip:
When solving matrix equations, always match corresponding elements carefully to ensure you solve for each variable correctly.

This problem involves solving for the variables x and y by performing element-wise addition on two matrices. Follow this step-by-step guide to solve the matrix equation. Suitable for students in Grades 10-12.

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