• know how to solve a linear equation in terms of one variable: one-step lesson, two-step lesson, multi-step lesson
• know how to solve systems using substitution
• know how to solve systems using elimination-by-addition

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: