A line with slope $-2$ passing through $(0,6)$ has equation $y=-2x+6$. At the intersection with $y=x$, the two expressions for $y$ are equal, so $x=-2x+6$. Adding $2x$ to both sides gives $3x=6$, so $x=2$.

Add the two equations in the system:

The $y$ terms cancel, so

. That is the equivalent equation in terms of $x$ only.

At the intersection, both equations have the same $y$-value. Since one line is $y=7$, substitute $7$ for $y$ in $y=2x+1$:

Subtract $1$ from both sides:

So $x=3$. Therefore, $m=3$.

Multiply the first equation by $2$:

This is exactly the second equation. That means both equations represent the same line, so every point that satisfies one equation also satisfies the other. Therefore, the system has infinitely many solutions.

Let the width be $w$ feet and the length be $l$ feet. The perimeter gives the equation

which simplifies to

The relationship between the dimensions gives

Substitute into the first equation: $w+9+w=27$, so $2w=18$ and $w=9$. Then $l=18$. The area is