First find the slope using the two given points:

So the line has equation $y=-x+b$. Substitute one of the given points, such as $(2,7)$:

so $b=9$. The equation is

Now check which point satisfies the equation. For $(0,9)$, $y=-0+9=9$, so this point is on the line.

Use the two given points to find the slope:

Since the line crosses the $y$-axis at $(0,-3)$, its $y$-intercept is $b=-3$. In slope-intercept form,

so the equation is

First find the slope:

Then use $y=ax+b$ with one of the points, such as $(2,11)$:

So

which gives

Since the line crosses the $y$-axis at $6$, its equation has the form $y=mx+6$. It also passes through $(4,-2)$, so substitute to find the slope: $-2=4m+6$. Then $4m=-8$, so $m=-2$. The equation is $y=-2x+6$. Rewriting in an equivalent form gives $2x+y=6$.

Let $m$ be the monthly fee and $s$ be the one-time sign-up fee. Then the two payments can be modeled by the system

Subtract the first equation from the second to eliminate $s$:

So,

The monthly fee is \12$.$