Let's break down the steps to solve this problem where we need to find the number of adults and students who attended the school play.
We have two key pieces of information:
1. The total number of people who attended the play is 950.
2. The total ticket sales amount is [tex]\( \$1200 \)[/tex], with adults paying [tex]\( \$2 \)[/tex] each and students paying [tex]\( \$1 \)[/tex] each.
Let's define:
- [tex]\( A \)[/tex] as the number of adults who attended.
- [tex]\( S \)[/tex] as the number of students who attended.
Using this information, we can set up the following two equations:
1. The total number of attendees:
[tex]\[ A + S = 950 \][/tex]
2. The total revenue from ticket sales:
[tex]\[ 2A + 1S = 1200 \][/tex]
We'll solve these equations simultaneously.
Below, I'll summarize the process to find the number of adults ([tex]\( A \)[/tex]) and students ([tex]\( S \)[/tex]):
1. Start with the first equation:
[tex]\[ A + S = 950 \][/tex]
2. Solve this equation for [tex]\( S \)[/tex]:
[tex]\[ S = 950 - A \][/tex]
3. Substitute [tex]\( S = 950 - A \)[/tex] into the second equation:
[tex]\[ 2A + 1(950 - A) = 1200 \][/tex]
4. Simplify the equation:
[tex]\[ 2A + 950 - A = 1200 \][/tex]
[tex]\[ A + 950 = 1200 \][/tex]
[tex]\[ A = 1200 - 950 \][/tex]
[tex]\[ A = 250 \][/tex]
5. Now, substitute [tex]\( A = 250 \)[/tex] back into the first equation to find [tex]\( S \)[/tex]:
[tex]\[ 250 + S = 950 \][/tex]
[tex]\[ S = 950 - 250 \][/tex]
[tex]\[ S = 700 \][/tex]
Therefore, the number of adults who attended the play is [tex]\( 250 \)[/tex], and the number of students who attended is [tex]\( 700 \)[/tex].
So, the completed equation is:
[tex]\[
\begin{array}{c}
2A + S = 1200 \\
A + S = 950
\end{array}
\][/tex]
This gives us the solution:
[tex]\[ A = 250 \][/tex]
[tex]\[ S = 700 \][/tex]
