Because $(2,1)$ is a solution, it must satisfy both equations. Substituting $x=2$ and $y=1$ gives

and

Simplify the second equation:

Now solve the system

Subtract the first equation from the second:

Then substitute into $2a+b=7$:

Therefore,

One equation already shows that $x+y=9$. Since the question asks for the value of $x+y$, no further solving is needed. The value is $9$.

First find the equation of the line through $(0,4)$ and $(2,0)$. Its slope is $\frac{0-4}{2-0}=-2$, so the equation is $y=-2x+4$. At the intersection, both expressions for $y$ are equal, so set $-2x+4=x+1$. Then $3=3x$, so $x=1$.

First solve the first equation for $y$:

Now substitute this expression for $y$ into the second equation, $4x+6y=36$:

That is exactly the equation in the correct answer. It is equivalent to the original system because it comes from replacing $y$ in one equation with an equal expression from the other equation.

Let $s$ be the number of student tickets and $a$ be the number of adult tickets. The situation gives the system

and

Substitute $s=180-a$ into the revenue equation:

Then

so

Subtract $1080$ from both sides to get

and divide by $4$:

So $80$ adult tickets were sold.