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Solve this question with explanation and find x and y 

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.

Solve for matrices X and Y given X + Y = [[7, 0], [2, 5]] and X - Y = [[3, 3], [0, 3]].

Learn how to find matrices X and Y given the matrix equations X + Y = [[7, 0], [2, 5]] and X - Y = [[3, 3], [0, 3]]. This solution involves using properties of matrix addition and subtraction to simplify the system of equations. Suitable for high school students in Grades 10-12.

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