Hello!
First, let's write some important information contained in the exercise:
Fees:
• $8.00 for children
,• $11.50 for adults
• On a certain day, ,321 people entered to park
,• The admission fees ,collected totaled $3,173.50
Knowing that we can write it as a system, look:
[tex]\begin{cases}8c+11.5a=3,173.50\text{ equation i)} \\ c+a=321\text{ equation ii)}\end{cases}[/tex]Let me explain the system:
Equation i) means that the total value obtained from the sale of tickets was $3,173.50.
Equation ii) means that the total of adults and children was 321 people.
I will isolate the variable c in equation ii), look:
[tex]\begin{gathered} c+a=321 \\ c=321-a \end{gathered}[/tex]So from now on, we're going to use C = 321 - a.
Now, let's replace the value of C in the equation i):
[tex]\begin{gathered} 8c+11.5a=3,173.50 \\ 8\cdot(321-a)+11.5a=3,173.50 \\ 2,568-8a+11.5a=3,173.50 \\ 2,568+3.5a=3,173.50 \\ 3.5a=3,173.50-2,568 \\ 3.5a=605.50 \\ a=\frac{605.50}{3.5} \\ a=173 \end{gathered}[/tex]Now we know the number of adults as 173, let's replace it instead A in equation ii):
[tex]\begin{gathered} c+a=321 \\ c+173=321 \\ c=321-173 \\ c=148 \end{gathered}[/tex]According to the reasoning above, 148 children and 173 adults were admitted.