Substitution

Question

If tickets to a movie cost$7.25 for adults and $5.50 for students. A group of friends purchased 8 tickets for $52.75. What would be an equation to represent this equation? How many adult tickets and student tickets were purchased?

Answer 1

Let a = the number of adults and s = the number of students

This is a systems of linear equation problem, basically 2 equations with 2 unknowns.  The two equations are:

a + s = 8  (representing the number of people going to the movie)

7.25a + 5.50s = 52.75 (representing the total cost to attend the movie)

Solving the first equation for a (subtracting s from both sides) gives us a = 8 - s

Substituting the first equation into the second gives us 7.25(8 - s) + 5.50s = 52.75  Now there is only one unknown

Distributing the 7.25, the equation becomes 58 - 7.25s + 5.50s = 52.75

Adding the "s" terms, the equation becomes 58 - 1.75s = 52.75

Subtracting 58 from both sides yields -1.75s = -5.25

Dividing both sides by -1.75 gives us s = 3  That solution is good because in a problem like this you can NOT have a "part" of a person!  The number MUST be a whole number

Plugging s = 3 back into the first equation of a + s = 8 gives us a = 5

Therefore, 5 adults and 3 students

Answer 2

The first step in solving a story problem is to identify your variables.  For this problem, I will use the following variables:

x = number of adult tickets

y = number of students tickets

Now, we need to right a number of equations from the problem that equals the number of variables (2 in this case).

Since the there were a total of 8 tickets purchased:

x + y = 8

Since the amount spent on adult tickets (7.25x) plus (+) the amount spent on student tickets (5.5y) totaled (=) 52.75, we get the equation:

7.25x + 5.5 y = 52.75

Now that we have two equations, we can use either substitution or elimination method to solve.  Since you labeled this substitution, that is the method that I will use

x + y = 8                                       Equation 1

x + y - y = 8 - y                             Subtract y from each side

x = 8 - y                                       Simplify

7.25x + 5.5y = 52.75                     Equation 2

7.25(8 - y) + 5.5y = 52.75              Substitute the value of x from equation 1 into equation 2

58 - 7.25y + 5.5y = 52.75               Distribute the 7.25

58 - 1.75y = 52.75                         Simplify

58 - 1.75y - 58 = 52.75 - 58            Subtract 58 from each side

-1.75y = -5.25                               Simplify

-1.75y/(-1.75) = -5.25/(-1.75)         Divide each side by -1.75

y = 3

Now that we have the value of y, we can plug it back into equation 1 to find x.

x = 8 - y              Equation 1 solved for x

x = 8 - 3              Substitution

x = 5

There were 5 adult tickets and 3 student tickets sold.