Question

1. a local drama production sold tickets for a play. adult tickets sold for $10 each and child tickets sold for $5 each. during the weekend, they collected $1750 from ticket sales. there was a total of 225 tickets sold.

a) write a system of equations to represent the total cost and another equation to represent the total number of people.
b) graph the equation of the total number of tickets sold and amount of money from sold tickets.
c) how many adult and children tickets were sold?
d) verify your solution.

Explanation:

Step1: Define variables

Let $x$ = number of children tickets, $y$ = number of adult tickets.

Step2: Write total tickets equation

Total tickets sold: $x + y = 225$

Step3: Write total revenue equation

Total sales: $5x + 10y = 1750$

Step4: Simplify revenue equation

Divide by 5: $x + 2y = 350$

Step5: Solve system via subtraction

Subtract total tickets eq from simplified revenue eq:
$$(x + 2y) - (x + y) = 350 - 225$$
$$y = 125$$

Step6: Find $x$ using total tickets eq

Substitute $y=125$ into $x + y = 225$:
$$x = 225 - 125 = 100$$

Step7: Verify total tickets

Check $x + y = 100 + 125 = 225$

Step8: Verify total revenue

Check $5(100) + 10(125) = 500 + 1250 = 1750$

Answer:

a) System of equations:
$x + y = 225$
$5x + 10y = 1750$
(where $x$ = number of children tickets, $y$ = number of adult tickets)

b) Graph details:

• For $x + y = 225$:
• When $x=0$, $y=225$; when $y=0$, $x=225$. Plot and connect these two points.
• For $5x + 10y = 1750$ (or $x + 2y = 350$):
• When $x=0$, $y=175$; when $y=0$, $x=350$. Plot and connect these two points.
• The lines intersect at $(100, 125)$.

c) 100 children tickets and 125 adult tickets were sold.

d) Verification:

• Total tickets: $100 + 125 = 225$, which matches the given total.
• Total revenue: $5(100) + 10(125) = 500 + 1250 = 1750$, which matches the given sales amount. The solution is valid.