At a county fair, adults' tickets sold for $5.50, senior citizens' tickets for $4.00, and children's tickets for $1.50. On the opening day, the number of children's and senior's tickets sold was 30 more than half of the number of adults' tickets sold. the number senior citizens' tickets sold was 5 more than four times the number of children's tickets. How many of each type of ticket were sold if the total receipts from the ticket sales were $14, 970?

let a = no. adult tickets

let s = no. senior tickets
let c = no. child tickets

c + s = 30 + 1/2 a
s = 5 + 4c
5.5a + 4s + 1.5c = 14970

I'll leave the solution to you. Substitution, Gaussian elimination, determinants, whatever.

To solve this problem, we can set up a system of equations.

Let's start by assigning variables to the unknown quantities:
Let A be the number of adult tickets sold.
Let S be the number of senior tickets sold.
Let C be the number of children's tickets sold.

From the given information, we can set up the following equations:
1) The total number of tickets sold is equal to the sum of adult tickets, senior tickets, and children's tickets:
A + S + C = Total number of tickets ----- (Equation 1)

2) The price of adult tickets is $5.50, senior tickets is $4.00, and children's tickets is $1.50:
5.50A + 4.00S + 1.50C = $14,970 ------ (Equation 2)

3) The number of children's and senior's tickets sold was 30 more than half of the number of adults' tickets sold:
C + S = (A/2) + 30 ------ (Equation 3)

4) The number of senior citizens' tickets sold was 5 more than four times the number of children's tickets:
S = 4C + 5 ------ (Equation 4)

Now, we can solve this system of equations to find the values of A, S, and C.

From Equation 3, we can express C in terms of A and S:
C = (A/2) + 30 - S

Substituting this into Equation 4, we get:
S = 4((A/2) + 30 - S) + 5
Simplify:
S = 2A + 60 - 4S + 5
5S = 2A + 65 - 60
5S = 2A + 5
2A = 5S - 5
2A = 5(S - 1) ----- (Equation 5)

Now, we'll substitute Equations 3 and 5 into Equation 1:
A + (5(S - 1)/2) + ((A/2) + 30 - S) = Total number of tickets
Simplify:
A + (5S - 5)/2 + A/2 + 30 - S = Total number of tickets
5A + 5S - 5 + A + 60 - 2S - 2A = Total number of tickets
4A + 3S + 55 = Total number of tickets
4A + 3S = Total number of tickets - 55 ----- (Equation 6)

Now we have two equations to work with:
2A = 5(S - 1) ------ (Equation 5)
4A + 3S = Total number of tickets - 55 ------ (Equation 6)

Using Equation 6, we can express A in terms of S:
4A = Total number of tickets - 3S - 55
A = (Total number of tickets - 3S - 55)/4

Substituting this into Equation 5, we get:
2((Total number of tickets - 3S - 55)/4) = 5(S - 1)
Multiply both sides by 4 to eliminate the fraction:
2(Total number of tickets - 3S - 55) = 20(S - 1)
2Total number of tickets - 6S - 110 = 20S - 20
2Total number of tickets = 6S + 20S - 20 + 110
2Total number of tickets = 26S + 90
Total number of tickets = (26S + 90)/2
Total number of tickets = 13S + 45

We now have an expression for the total number of tickets in terms of S.

Since the total number of tickets must be a whole number, we can set 13S + 45 equal to a factor or multiple of 13. Let's set it equal to 13.

13S + 45 = 13
13S = 13 - 45
13S = -32
S = -32/13

However, S cannot be negative, so there is no valid solution in this case.

Therefore, there is no combination of tickets that satisfies all the given conditions and the total receipt amount of $14,970.