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The given problem is a matrix equation involving two matrices that are set equal to each other. The task is to find the values of \( x \) and \( y \) that satisfy the matrix equation. The matrices are:

\[
\begin{pmatrix}
-8 & 7 & 1 \\
x & x+2 & -2
\end{pmatrix}
-
\begin{pmatrix}
2y + 1 & 6 & 6 \\
1 & 5 & y
\end{pmatrix}
=
\begin{pmatrix}
-7 & -5 & 6 \\
-7 & 2 & 6
\end{pmatrix}
\]

### Step 1: Write out the matrix subtraction

Let's subtract the second matrix from the first matrix element-wise:

\[
\begin{pmatrix}
-8 & 7 & 1 \\
x & x+2 & -2
\end{pmatrix}
-
\begin{pmatrix}
2y + 1 & 6 & 6 \\
1 & 5 & y
\end{pmatrix}
=
\begin{pmatrix}
(-8 - (2y + 1)) & (7 - 6) & (1 - 6) \\
(x - 1) & ((x+2) - 5) & (-2 - y)
\end{pmatrix}
\]

### Step 2: Simplify each element of the resulting matrix

Simplifying each term:

\[
\begin{pmatrix}
-8 - (2y + 1) & 7 - 6 & 1 - 6 \\
x - 1 & (x + 2 - 5) & -2 - y
\end{pmatrix}
=
\begin{pmatrix}
-9 - 2y & 1 & -5 \\
x - 1 & x - 3 & -2 - y
\end{pmatrix}
\]

### Step 3: Set up the element-wise equations

We now compare this resulting matrix to the matrix on the right-hand side:

\[
\begin{pmatrix}
-9 - 2y & 1 & -5 \\
x - 1 & x - 3 & -2 - y
\end{pmatrix}
=
\begin{pmatrix}
-7 & -5 & 6 \\
-7 & 2 & 6
\end{pmatrix}
\]

This gives us the following system of equations by equating corresponding elements:

1. \( -9 - 2y = -7 \)
2. \( 1 = -5 \) (which results in a contradiction)
3. \( -5 = 6 \) (which also results in a contradiction)
4. \( x - 1 = -7 \)
5. \( x - 3 = 2 \)
6. \( -2 - y = 6 \)

### Step 4: Solve for \( y \) and \( x \)

Let's solve the valid equations:

- From equation 1: \( -9 - 2y = -7 \)

\[
-2y = -7 + 9 = 2 \quad \Rightarrow \quad y = -1
\]

- From equation 4: \( x - 1 = -7 \)

\[
x = -7 + 1 = -6
\]

- From equation 5: \( x - 3 = 2 \)

\[
x = 2 + 3 = 5
\]

There seems to be an inconsistency between these equations for \( x \). This suggests that either there is no solution or the system is overdetermined. However, for the purpose of this exercise, it seems we only needed to solve for \( x \) and \( y \), assuming one of the two solutions for \( x \) is valid depending on the context.

Would you like more details or further clarification?

Here are some additional questions to explore:
1. How do you handle inconsistent systems of equations?
2. What conditions make two matrices equal?
3. How can matrix subtraction be applied in solving systems of equations?
4. What other methods can solve for unknowns in matrix equations?
5. What does it mean when a system of equations has no solution?

**Tip:** Always check each step when solving systems of equations to ensure that contradictions don't go unnoticed.

Solve for x and y in the given matrix equation:
\[
\begin{pmatrix}
-8 & 7 & 1 \\
x & x+2 & -2
\end{pmatrix}
-
\begin{pmatrix}
2y + 1 & 6 & 6 \\
1 & 5 & y
\end{pmatrix}
=
\begin{pmatrix}
-7 & -5 & 6 \\
-7 & 2 & 6
\end{pmatrix}
\]

Find the values of x and y that satisfy the given matrix equation using matrix subtraction. This problem involves comparing elements and solving a system of linear equations. Suitable for students in Grades 11-12.

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