To solve the problem, let's go through the details presented in the image. The question asks us to find matrices $X$ and $Y$ given two conditions:

1. $X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$
2. $X - Y = \begin{bmatrix} 3 & 3 \\ 0 & 3 \end{bmatrix}$

To solve for $X$ and $Y$, we can use these two equations to eliminate one of the matrices by adding or subtracting them.

Step 1: Add the Two Equations

$(X + Y) + (X - Y) = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix} + \begin{bmatrix} 3 & 3 \\ 0 & 3 \end{bmatrix}$

This simplifies to:

$2X = \begin{bmatrix} 10 & 3 \\ 2 & 8 \end{bmatrix}$

Now, divide by 2 to solve for $X$:

$X = \begin{bmatrix} 5 & 1.5 \\ 1 & 4 \end{bmatrix}$

Step 2: Substitute $X$ into One of the Original Equations to Find $Y$

Using the first equation $X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$, substitute the values of $X$:

$\begin{bmatrix} 5 & 1.5 \\ 1 & 4 \end{bmatrix} + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$

Now, solve for $Y$ by subtracting $X$ from both sides:

$Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix} - \begin{bmatrix} 5 & 1.5 \\ 1 & 4 \end{bmatrix}$

This simplifies to:

$Y = \begin{bmatrix} 2 & -1.5 \\ 1 & 1 \end{bmatrix}$

Final Solution

The matrices $X$ and $Y$ are:

$X = \begin{bmatrix} 5 & 1.5 \\ 1 & 4 \end{bmatrix}$ $Y = \begin{bmatrix} 2 & -1.5 \\ 1 & 1 \end{bmatrix}$

Would you like further clarification on any step, or do you have any other questions?

Here are some related questions for further exploration:

1. How would the solution change if we were given different matrices for $X + Y$ and $X - Y$?
2. What if we had $3X + 2Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$ and $X - Y = \begin{bmatrix} 3 & 3 \\ 0 & 3 \end{bmatrix}$?
3. How can we verify that our solution for $X$ and $Y$ is correct?
4. What properties of matrix addition and subtraction are used in this solution?
5. How would we solve this problem if $X$ and $Y$ were 3x3 matrices?

Tip: When solving matrix equations, adding and subtracting the equations can often simplify the process by eliminating one of the unknowns.