In order to solve problems involving systems of linear equations, it is necessary to

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 6

**s**. 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 9

**n**. Altogether, the movie theater collected $4974. We need to show that the money collected from senior citizens (6

**s**) plus the money collected from non-senior citizens (9

**n**) 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: