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 people from each group attended the show ?

To solve this problem, we will break it down into steps:

Step 1: Define the variables
Let's define the following variables:
C = Number of children
S = Number of students
A = Number of adults

Step 2: Set up equations based on the given information
According to the problem, the seating capacity of the theater is 900. Therefore, the total number of people attending the show is C + S + A, and this should be equal to 900:
C + S + A = 900 (Equation 1)

The problem also states that there were half as many adults as children and students combined:
A = (C + S)/2 (Equation 2)

Step 3: Set up an equation for calculating the total receipts
The total receipts from the show is given as $5600. Since children pay $4, students pay $6, and adults pay $8, we can set up the following equation:
4C + 6S + 8A = 5600 (Equation 3)

Step 4: Solve the system of equations
To find the values of C, S, and A, we need to solve the system of equations formed in Steps 2 and 3.

One way to solve this system is by substitution:

Substitute equation 2 into equation 3:
4C + 6S + 8((C + S)/2) = 5600

Simplify the equation:
4C + 6S + 4C + 4S = 5600
8C + 10S = 5600 (Equation 4)

Now, substitute equation 2 into equation 1:
C + S + ((C + S)/2) = 900

Simplify the equation:
2C + 2S + C + S = 1800
3C + 3S = 1800 (Equation 5)

Divide equation 4 by 2 to simplify:
4C + 5S = 2800 (Equation 6)

Now, we have a system of equations:
3C + 3S = 1800 (Equation 5)
4C + 5S = 2800 (Equation 6)

Solving this system of equations, we find C = 300 and S = 200.

Step 5: Calculate the value for A
Substitute the values of C and S back into equation 2 to find A:
A = (C + S)/2
A = (300 + 200)/2
A = 500/2
A = 250

Therefore, there were 300 children, 200 students, and 250 adults who attended the show.

so, what do you know?

c+s+a = 900
a = (c+s)/2
4c+6s+8a = 5600

3 equations in 3 unknowns. Just solve them for c,s,a