2011 AMC 10A Problems/Problem 12

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Problem 12

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?

$\text{(A)}\,13 \qquad\text{(B)}\,14 \qquad\text{(C)}\,15 \qquad\text{(D)}\,16 \qquad\text{(E)}\,17$

Solution

Suppose there were $x$ three-point shots, $y$ two-point shots, and $z$ one-point shots. Then we get the following system of equations:

\[\begin{array}
3x=2y\\ z=y+1\\ 3x+2y+z=61
\end{array}\] (Error compiling LaTeX. ! LaTeX Error: Illegal character in array arg.)

The value we are looking for is $z$, which is easily found to be $z=\boxed{13 \ \mathbf{(A)}}$.

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