Answer :
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]
[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]
