Question

Gavin is a Highschool bascketball player. In a particular game, he made some free throws and some three point shots. Gavin made a total of 13 shots altogether and scored a total of 25 points. Determine the number of free throws Gavin made and the number of three point shots he made.

Answer 1

Let \( x \) be the number of free throws Gavin made, and \( y \) be the number of three-point shots he made. We can set up the following system of equations based on the problem statement:

1. The total number of shots made:
\[
x + y = 13
\]

2. The total points scored (free throws are worth 1 point, and three-point shots are worth 3 points):
\[
x + 3y = 25
\]

Now we can solve this system of equations. We can start by solving the first equation for \( x \):
\[
x = 13 - y
\]

Next, we substitute \( x \) in the second equation:
\[
(13 - y) + 3y = 25
\]

Simplifying this:
\[
13 - y + 3y = 25
\]
\[
13 + 2y = 25
\]

Now, isolate \( y \):
\[
2y = 25 - 13
\]
\[
2y = 12
\]
\[
y = 6
\]

Now that we have \( y = 6 \), we can find \( x \) using the first equation:
\[
x + 6 = 13
\]
\[
x = 13 - 6
\]
\[
x = 7
\]

Thus, the number of free throws Gavin made is \( 7 \) and the number of three-point shots he made is \( 6 \).

To summarize, Gavin made:
- Free throws: \( 7 \)
- Three-point shots: \( 6 \)

We can verify our solution:
- The total number of shots is \( 7 + 6 = 13 \).
- The total points scored is \( 7 \cdot 1 + 6 \cdot 3 = 7 + 18 = 25 \).

Both conditions are satisfied, confirming that our solution is correct.