Question

A theater has a seating capacity of 900 and charges $4 for children, $6 for students, and $8 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $5600. How many children attended the show?

Answer 1

a = Adults amount
c = Children amount
s = Students amount

the theater has a capacity of 900 total, that day there was full attendance, so all 900 seats were used, so there were 900 adults and students and children
thus
\(\bf a + c + s = 900\)

the cost of each "a" is $8, "c" is $4 and "s" is $6, their total receipt was $5600
thus

\(\bf 8a + 4c + 6s = 5600\)

Answer 2

Thank you @jdoe0001 is there anything else I should do next?

Answer 3

we know that, there were half-adults as "children and students combined"
so a = 1/2 (c + s)
thus \(\bf a + c + s = 900 \implies \left( \cfrac{ c+s }{ 2 } \right)+ c + s = 900\\
\textit{multiplying both sides by "2" to get rid of the denominator}\\
2 \times \left(\cfrac{ c+s }{ 2 }+ c + s\right) = 900 \times 2
\implies c+s+2c+2s = 1800\\\quad \\
3c = 1800-3s \implies c = \cfrac{ 1800-3s }{ 3 } \implies c = 600 -s
\)

Answer 4

so we know what c = 600 -s
or we can also say that s = 600 -c

so, let's use that in our second equation
\(\bf 8a + 4c + 6s = 5600\qquad \qquad a = \cfrac{ c+s }{ 2 }\qquad \qquad s = 600-c\)

if you use those values in the equation, how would it look like?