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------------------------------------------------ Solve for [tex]$x$[/tex] and [tex]$y$[/tex].

[tex]
\[
\begin{array}{c}
\left[
\begin{array}{ccc}
x+2 & 6 & -3 \\
9 & 18 & -6 \\
9 & -2 & y+2
\end{array}
\right]
=
\left[
\begin{array}{ccc}
2x+6 & 6 & -3 \\
9 & 18 & -6 \\
9 & -2 & x
\end{array}
\right]
\end{array}
\]
[/tex]

[tex]x = \square[/tex]

[tex]y = \square[/tex]

To solve the matrix equation

[tex]$$
\begin{bmatrix}
x+2 & 6 & -3 \\
9 & 18 & -6 \\
9 & -2 & y+2
\end{bmatrix}
=
\begin{bmatrix}
2x+6 & 6 & -3 \\
9 & 18 & -6 \\
9 & -2 & x
\end{bmatrix},
$$[/tex]

we equate the corresponding entries from both matrices.

First, compare the [tex]$(1,1)$[/tex] entries:

[tex]$$
x+2 = 2x+6.
$$[/tex]

Subtract [tex]$x$[/tex] from both sides to isolate [tex]$x$[/tex]:

[tex]$$
2 = x+6.
$$[/tex]

Subtract [tex]$6$[/tex] from both sides:

[tex]$$
x = -4.
$$[/tex]

Next, compare the [tex]$(3,3)$[/tex] entries:

[tex]$$
y+2 = x.
$$[/tex]

Since we found [tex]$x = -4$[/tex], substitute:

[tex]$$
y+2 = -4.
$$[/tex]

Subtract [tex]$2$[/tex] from both sides to solve for [tex]$y$[/tex]:

[tex]$$
y = -6.
$$[/tex]

Thus, the solutions are:

[tex]$$
x = -4 \quad \text{and} \quad y = -6.
$$[/tex]