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.