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To solve the given problem, we need to find the values of \(x\) and \(y\) based on the equations provided. Here's the detailed solution:

### Step 1: Information given
- \( A^T = \begin{pmatrix} 1 & 2 & 2x+1 \\ 1 & 1 & 2 \\ 0 & -3 & 1 \end{pmatrix} \)
- \( B^T = \begin{pmatrix} 1 & 2 & 2 \\ -2 & 1 & 0 \\ 1 & 3 & 4 \end{pmatrix} \)
- \( (A + B)^T = \begin{pmatrix} 2 & 4 & 6-x \\ -1 & 4 & 2 \\ 1 & 0 & 3y-1 \end{pmatrix} \)

### Step 2: Compute \( (A + B)^T \)
The transpose of the sum of two matrices is the sum of their transposes. Hence:
\[
(A + B)^T = A^T + B^T
\]

Adding \( A^T \) and \( B^T \):
\[
A^T + B^T = \begin{pmatrix} 1 & 2 & 2x+1 \\ 1 & 1 & 2 \\ 0 & -3 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 2 \\ -2 & 1 & 0 \\ 1 & 3 & 4 \end{pmatrix}
\]

Perform element-wise addition:
\[
A^T + B^T = \begin{pmatrix} 1+1 & 2+2 & (2x+1)+2 \\ 1+(-2) & 1+1 & 2+0 \\ 0+1 & -3+3 & 1+4 \end{pmatrix}
= \begin{pmatrix} 2 & 4 & 2x+3 \\ -1 & 2 & 2 \\ 1 & 0 & 5 \end{pmatrix}
\]

### Step 3: Equate with the given \( (A + B)^T \)
The given matrix for \( (A + B)^T \) is:
\[
(A + B)^T = \begin{pmatrix} 2 & 4 & 6-x \\ -1 & 4 & 2 \\ 1 & 0 & 3y-1 \end{pmatrix}
\]

Comparing elements of the two matrices:
1. From the \((1, 3)\) element: \( 2x + 3 = 6 - x \)
2. From the \((2, 2)\) element: \( 2 = 4 \) (This matches, so no new information.)
3. From the \((3, 3)\) element: \( 5 = 3y - 1 \)

### Step 4: Solve for \(x\)
From \( 2x + 3 = 6 - x \):
\[
2x + x = 6 - 3 \implies 3x = 3 \implies x = 1
\]

### Step 5: Solve for \(y\)
From \( 5 = 3y - 1 \):
\[
5 + 1 = 3y \implies 6 = 3y \implies y = 2
\]

### Final Answer:
\[
x = 1, \quad y = 2
\]

Would you like a detailed explanation of how the matrix addition works or any further clarification? 

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### Additional Questions:
1. What is the transpose of a matrix, and how does it affect matrix addition?
2. Can this problem be solved if one element in \( (A + B)^T \) is missing? Why or why not?
3. How do you compute the determinant of \( A^T + B^T \)?
4. What are the properties of the transpose operation in relation to matrix multiplication?
5. How do you verify if two matrices are equal?

### Tip:
Always double-check your arithmetic when solving matrix problems, as small errors can lead to incorrect results!

If \( A^T = \begin{pmatrix} 1 & 2 & 2x+1 \\ 1 & 1 & 2 \\ 0 & -3 & 1 \end{pmatrix}, \)
\( B^T = \begin{pmatrix} 1 & 2 & 2 \\ -2 & 1 & 0 \\ 1 & 3 & 4 \end{pmatrix}, \)
and \( (A + B)^T = \begin{pmatrix} 2 & 4 & 6-x \\ -1 & 4 & 2 \\ 1 & 0 & 3y-1 \end{pmatrix}, \)
find the values of \( x \) and \( y \).

Learn how to solve for x and y in a matrix equation involving transposed matrices A^T and B^T. This step-by-step guide explains matrix addition, transposition, and solving linear equations. Suitable for high school students (Grade 11-12).

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