Objectives 

• Identifying the Relevant Variables

• Constructing Equations 

• Solving Equations and Checking Solutions

Examples

• Example 1

• Example 2

• Example 3

• Example 4

The UNC Basketball team scores a total of 117 points in its first game. The team makes twice as many field goals (2-points each) as free throws (1 point each), and twice as many free throws as 3-pointers. How many of each type of shot does UNC make?

Hint: Let  x = the number of 3-pointers made. 
 
 

Solution
    Our goal is to construct a one-variable equation which 
    matches the given information.

    Step 1: Identify the desired variable.

        The hint tells us to use the variable "x" to represent the
        number of 3-pointers made.

        That is,

x = the number of 3-pointers made.

    Step 2: Identify the known relationships in the problem.

        The problem statement tells us that UNC made twice as
        many free throws as 3-pointers. 

     We can translate this fact as follows:

the number of free throws made = 2(x) = 2x

        The problem statement also tells us that UNC made twice
        as many field goals as free throws. 

     We can translate this fact as follows:

the number of field goals made = 2(2(x)) = 4x

       We will also need the formula:

 (# made)(point value) = points earned 

        We can construct a table to help us organize the given
        information.
 

	# made	point value	points earned
3-pointers	x	3	(x)(3)
free throws	2x	1	(2x)(1)
field goals	4x	2	(4x)(2)

Step 3: Construct a one variable equation which fits the known
              relationships in the problem.

        The problem statement tells us that UNC scored a total of 
        117 points.

        Using the entires from the "points earned" column in our
        table, we have the equation:

Total Points = 117 

2x + 8x + 3x = 117

Step 4: Solve the equation.

        Solving for x, we get

13x = 117

 x = (117)/(13) = 9 

Step 5: Use the solution from Step 4 to answer the question(s)
              posed in the original problem. 

Plugging the value x = 9 into our table, we see that  UNC made  9 three-pointers, 18 free-throws and 36 field-goals. 


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Special Note: Be sure to check that your final answers "make sense" in the context of the original problem. 

In this problem, for example, our equation needed to yield a positive whole number for the value of "x". 

That is, it wouldn't "make sense" for UNC to have made - 7 free throws or 5.6 field goals.