You can buy 1 child's ticket, 1 adult's ticket and 1 senior citizen's ticket for a minor league baseball game for $29. Tickets for adults cost 3 times as much as tickets for children. A family of 3 children, 2 adults, and 2 senior citizens pays a total of $63 for tickets to the game. Find the prices of each type of ticket.
To solve this problem, we can set up a system of equations based on the given information. Let's denote the cost of a child's ticket as "c," the cost of an adult's ticket as "a," and the cost of a senior citizen's ticket as "s."
From the first statement, we know that 1 child's ticket + 1 adult's ticket + 1 senior citizen's ticket equals $29:
c + a + s = 29 ....(1)
The second statement tells us that a family with 3 children, 2 adults, and 2 senior citizens paid a total of $63 for tickets:
3c + 2a + 2s = 63 ....(2)
Since we have two equations and three unknowns, we need to find an additional equation.
From the given information that "Tickets for adults cost 3 times as much as tickets for children," we can write:
a = 3c ....(3)
Now, we have a system of three equations:
c + a + s = 29 ....(1)
3c + 2a + 2s = 63 ....(2)
a = 3c ....(3)
We can solve this system of equations to find the values of c, a, and s.
First, substitute equation (3) into equations (1) and (2):
c + 3c + s = 29 ....(1')
3c + 2(3c) + 2s = 63 ....(2')
Simplifying equation (1') and equation (2'):
4c + s = 29 ....(4)
9c + 2s = 63 ....(5)
We now have a system of two equations with two unknowns (equations 4 and 5).
To solve this system, we can use the method of substitution or elimination.
Let's use the substitution method to solve equations (4) and (5).
From equation (4), we can express s in terms of c:
s = 29 - 4c ....(6)
Substitute equation (6) into equation (5):
9c + 2(29 - 4c) = 63
Expand and simplify the equation:
9c + 58 - 8c = 63
c + 58 = 63
c = 63 - 58
c = 5
Now that we have the value of c, let's substitute it back into equation (6) to find the value of s:
s = 29 - 4(5)
s = 29 - 20
s = 9
Finally, substitute the values of c and s into equation (3) to find the value of a:
a = 3(5)
a = 15
Therefore, the prices of each type of ticket are:
Child's ticket (c) = $5
Adult's ticket (a) = $15
Senior citizen's ticket (s) = $9
c + a + s = 29 (A)
a = 3c (B)
3c + 2a + 2s = 63 (C)
multiply (A) by two, and subtract from (C)
... this eliminates a and s
substitute back with the value of c to find a and s