First find the slope of line $k$ using its two points:

A line perpendicular to a line with slope $2$ has slope $-\frac{1}{2}$. So the desired line has the form

Since it passes through $(4,5)$, substitute $x=4$ and $y=5$:

So $b=7$, and the equation is

Let $a$ be the number of adult tickets and $s$ be the number of student tickets. The total number of tickets gives $a+s=120$. The total revenue gives $18a+12s=1800$. Divide the second equation by $6$ to get $3a+2s=300$. Substitute $a=120-s$ into that equation:

So, $60$ student tickets were sold.

For a rectangle, the perimeter is $2(\text{length}+\text{width})$. The width is $w$, and the length is $2w+3$ because it is $3$ feet more than twice the width. Since the perimeter is at most $54$, the inequality is

That matches the correct expression.

Let $s$ be the one-time sign-up fee. Then the total cost after 3 months is $s+3m=47$, and the total cost after 7 months is $s+7m=83$. To eliminate the sign-up fee, solve each equation for $s$:

Since both expressions equal the same sign-up fee, set them equal:

So this is the correct equation.

Distribute each term first:

Then combine like terms:

So the expression is equivalent to