A play is performed for a crowd of 400 people. Adult tickets cost $22 each, student tickets cost $15 each, and tickets for children cost $13.50 each. The revenue for the concert is $7840. There are 40 more children at the concert than students. How many of each type of ticket are sold?

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carlosego carlosego · Answer 1 · 2018-10-19T02:31:22+02:00

Answer:

280 adult tickets, 40 student tickets and 80 children tickets were sold.

Step-by-step explanation:

To solve this problem let's call:

x = number of tickets for adults

y = number of tickets for students

z = number of tickets for children

The income for the concert is $ 7840

Then we can raise the following equations according to the given conditions:

[tex]x+y+z = 400[/tex] (i)

[tex]22x+15y +13.50z = 7840[/tex] (ii)

There are 40 more children in the concert than students:

So:

[tex]z=40+y[/tex] (iii)

We then have 3 equations and 3 unknowns:

To solve the system we multiply the equation (i) by -22 and add it to the equation (ii)

So:

[tex]-22x-22y-22z=-8800[/tex]

+

[tex]22x+15y +13.5z =7840 [/tex]

[tex]-7y-8.5z = -960[/tex] (iv)

Now we multiply (iii) by 7 and add it to (iv)

[tex]7y-7z = -280 [/tex]

+

[tex]-7y-8.5z = -960[/tex]

[tex]-15.5z = -1240 [/tex]

[tex]z = 80[/tex] (v)

We substitute (v) in (iv) and we have

[tex]y = 40[/tex] (vi)

We substitute (v) and (vi) into (i) and we have:

[tex]x =280[/tex]

280 adult tickets, 40 student tickets and 80 children tickets were sold.