SAT Question d7bf55e1: Systems of Linear Equations answer and explanation with Desmos solution

Let $a$ be the number of adult tickets sold and $c$ be the number of child tickets sold. Since the theater sold a total of $30$ tickets, $a+c=30$. The price per adult ticket is $$12$, and the price per child ticket is $$8$. Since the theater received a total of $$300$ for the $30$ tickets sold, it follows that $12a+8c=300$. To eliminate $c$, the first equation can be multiplied by $8$ and then subtracted from the second equation: $12a+8c=300$ and $-8a-8c=-240$ gives $4a+0c=60$. Because the question asks for the amount spent on adult tickets, which is $12a$ dollars, the resulting equation can be multiplied by $3$ to give $3(4a)=3(60)=180$. Therefore, $$180$ was spent on adult tickets.

Alternate approach: If all the $30$ tickets sold were child tickets, their total price would be $30(8)=240$ dollars. Since the actual total price of the $30$ tickets was $$300$, the extra $$60$ indicates that a certain number of adult tickets, $a$, were sold. Since the price of each adult ticket is $$4$ more than each child ticket, $4a=60$, and it follows that $12a=180$.