Linear System Application

Question

A movie theater charges $ 10 for adults and $ 7 for seniors. On a particular day when 345 people paid an admission, the total receipts were $ 2535. Round your answers to the nearest integers.

Answer 1

I assume you want to find how many adults and how many seniors there were.

 

A = number of adults

 

S = number of seniors

 

A + S = 345

 

A*($10) + S*($7) = $2535

 

The two equations you must solve are:

 

(i)  A + S = 345

(ii) 10A + 7S = 2535

 

From (i) get A = 345 - S and substitute into (ii) to find S. 

 

10(345 - S) + 7S = 2535

 

3450 - 10S + 7S = 2535

 

-3S = -915

 

S = 915/3 = 305

 

A = 345 - S = 345 - 305 = 40

 

There were 305 seniors and 40 adults.

 

Test the solution:

 

Is A + S = 345?

A + S = 40 + 305 = 345  [good]

 

Is 10A + 7S = 2535?

10A + 7S = 10(40) + 7(305) = 400 + 2135 = 2535  [good]

 

Answer 2

Let x = number of adults

      y = number of seniors

 

"345 people paid"   means     x+y = 345       [eq1]

 

"A movie theater charges $ 10 for adults and $ 7 for seniors."    means

    note: money for a group = ($ per person)*(number of people)

       charges for adults   +    charges for seniors    =    total charges

              $10x              +            $7y                 =         $2535             [eq2]

 

"Round your answers to the nearest integers."   Note: the answers must be integers (whole people) and should not need rounded.

 

Solve for one variable in terms of the other:

    x+y = 345    [eq1 again]

    x = 345 - y

 

Now, substitute that into the other equation:

   10x + 7y = 2535      [eq2 again]

   10(345 - y) + 7y = 2535               [substitute for x]

    3450 - 10y + 7y = 2535             [distribute]

        -3y = -915                         [collect terms; subtract 3450 from both sides]

        y = 305                     [divide both sides by (-3)]

 

Replace that for y in either equation (which is easier?):

       x = 345 - y             [from eq1]

       x = 345 - 305

       x = 40

 

Check:

  Is  10(40) + 7(305) = 2535   ?

           400 + 2135 = 2535   ?

              2535 = 2535   ?

 

Note: There must have been some cool nostalgic flicks showing !