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Word Problems: Ticket Sales
In order to solve problems involving systems of linear equations, it is necessary to
A typical problem involving systems of equations involves finding the ordered pair of numbers that simultaneously satisfy each equation in the system. The ordered pair will be the solution of the system.
An example of this type of problem is: A movie theater sells tickets for $9.00 each. Senior citizens receive a discount of $3.00. One evening the theater sold 636 tickets and took in $4974 in revenue. How many tickets were sold to senior citizens? How many were sold to “moviegoers” who were not senior citizens?
First we need to set up a system of two equations. The equations will be linear. One of the two will involve the number of people who attended the movie. There were a total of 636 people who attended the movie on the given day. There were senior citizens and non-senior citizens. Therefore, we need to assign two variables. One will represent senior citizens and the other will represent non-senior citizens. Let s represent senior citizens and n represent non-senior citizens. We need to show that the number of senior citizens (s) plus the number of non-senior citizens (n) equals the total number of people attending the movie (636). So we write:
The second equation will represent the amount of money collected for each ticket sold. Non-senior citizens are charged $9.00 for each ticket they purchase. Senior citizens get a $3.00 discount. Therefore senior citizens are charged $6.00 for each ticket they purchase (). Since $6.00 is received by the theater for each ticket sold to a senior citizen, and since senior citizens are represented with an s, we multiply 6 time s and obtain 6s. Likewise, since $9.00 is received by the theater for each ticket sold to a non-senior citizen, and since non-senior citizens are represented with an n, we multiply 9 times n and obtain 9n. Altogether, the movie theater collected $4974. We need to show that the money collected from senior citizens (6s) plus the money collected from non-senior citizens (9n) equals the total amount of money collected by the theater during the day described in the problem ($4974). So we write:
Now we have our system of two equations.
We are ready to solve the system for the ordered pair of numbers that represent the solution to the system. We will solve the system using the elimination-by-addition method. First we multiply the top row by and obtain:
Then we add the two equations to obtain:
Now we solve for s:
We have determined that 250 tickets were sold to senior citizens. Substituting 250 for s in the first equation of our original system of equations, we obtain:
By adding to each side of the equation, we obtain:
We have now determined that 386 tickets were sold to non-senior citizens. The ordered pair of numbers that represent the solution to our system of equations is the following:

Let's Practice

Question #1
AudioAt a high school championship basketball game 1200 tickets were sold. Student tickets cost $1.50 each and adult tickets cost $5.00 each. The total revenue collected for the game was $3200. How many student tickets were sold? How many adult tickets were sold?

Question #2
AudioA small town is hosting a play as a fundraising project for a family in the town. Individuals may purchase tickets for $26 each. As a special incentive to attract couples, individuals may purchase a “couple package” for $32. [Note: The “couple package” allows 2 people to attend the play for the price of $32.] On the first day of sales 23 tickets were sold and $694 was collected. How many tickets were sold to individuals for $26 each? How many people were convinced to buy the “couple package”?

Try These
Question #1
AudioThe treasurer of the student body at a college reported that the receipts from a recent concert totaled $916. Furthermore, he announced that 560 people had attended the concert. Students were charged $1.25 each for admission to the concert, and adults were charged $2.25 each. How many adults attended the concert?

Question #2
AudioFive hundred tickets were sold for a Saturday evening performance of a play. The tickets cost $7.50 for adults and $4.00 for children. A total of $3312.50 was received for all the tickets sold that Saturday evening. How many adults attended the play?

As you can see, this type of problem requires carefully setting up two equations with two unknown values. Following the writing of the two equations, you must carefully eliminate one of the variables and solve for the other. Then upon completion of the problem, you must substitute carefully into the two equations to check your answers.

D Saye

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