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