We have 3 variables to solve for. We will start by writing two equations in three variables but, with the given information, will eventually have those equations written in 2 variables. The strategy is to set up equations from the given information and solve the equations simultaneously. Let's assign variables...
c= #children total
s= #students total
a= #adults total
c+s+a=900.............Eq1, total attendance, given
4c+6s+8a=5600....Eq2, total receipts, given
a=(1/2)(c+s)
=(1/2)c+(1/2)s.....Eq3, #adults total, given
Let's substitute Eq3 into each Eqs1,2 to get two equations in two variables...
c+s+(½c+½s)=900.........Eq1, substitute for"s"
1.5c+1.5s=900.......combine like terms
3c+3s=1800.....Eq4, multiple equation
by 2 to eliminate fraction
4c+6s+8(½c+½s)=5600...Eq2, substitute for "s"
4c+6s+(4c+4s)=5600...distribute 8
8c+10s=5600...Eq5, combine like terms
Let's solve newly arrived at Eq4,5 simultaneuously...
3c+3s=1800......Eq4
8c+10s=5600......Eq5
-24c-24s=-14400...Eq4 multiplied by (-8)
24c+30s=16800...Eq5 multiplied by 3
------------------------
6s=2400.....add all terms vertically
∴ s=400.......divide both sides by 6
Substitute value back into Eq4, solve for "c"...
3c+3s=1800...Eq4
3c+3(400)=1800...substitute for"s"
3c+1200=1800
3c=600.....subtract 1200, both sides
∴ c=200....divide both sides by 3
Substitute values back into Eq3, solve for "a"...
a=(1/2)(c+s).............Eq3
=(1/2)(200+400)...substitute for "c,s"
=(1/2)(600)
∴ a=300.....................simplify
We now have our three variables solved. Let's substitute those values back into original Eqs1,2 to check against the problem requirements...
c+s+a=900.......Eq1
200+400+300=900....substitute for "c,s,a"
900=900......true, √check
4c+6s+8a=5600.....Eq2
4(200)+6(400)+8(300)=5600...substitute for
"c,s,a"
800+2400+2400=5600...simplify
5600=5600......true, √check
So our calculated values are correct. Always check your solutions and work!
In summary, we have...
200 children, 400 students, and 300 adults in attendance at the theater for the screening.
