Systems of equations. help!!!!

Question

The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,550 every night to cover all expenses. Let x represent the number of adult tickets sold at $7.50. Let y represent the number of student tickets sold at $4.50 each. If all 500 seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly $2,550? At one performance there were two times as many student tickets sold as adult tickets. If there were 300 tickets sold at that performance, how much below the goal of $2,550 did ticket sales fall?

Answer 1

Hi Aladdin,

 

To answer the first question, we have these two equations:

 

x + y = 500 and 7.5x + 4.5y = 2550

 

We can use substitution to solve for x and then plug it back into any equation and find y.

 

y = 500 - x

 

7.5x + 4.5 (500 - x) = 2550

 

7.5x + 2250 - 4.5x = 2550

 

3x = 300

 

x = 100 adults

 

100 + y = 500 → y = 400 students

 

To answer your second question, we have two new equations:

 

2x = y (This may be confusing, but since there were more student tickets sold, it is the larger number.)

x + y = 300

 

We can rewrite the first equation and add them together:

 

2x - y = 0

x + y = 300

 

3x = 300

 

x = 100 adults

 

100 + y = 300  y = 200 students

 

The money earned from these 300 people is 100 * 7.5 + 200 * 4.5 = 750 + 900 = $1650.

 

$2550 - $1650 = $900

 

The 300 tickets sold had a $900 fall in ticket sales.

 

Hope this helps!