2014 AMC 10A Problems/Problem 21

Problem

Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$ -axis at the same point. What is the sum of all possible $x$ -coordinates of these points of intersection?

$\textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-8}$

Solution

Note that $y=ax+5$ intersects the $x-$ axis at $(-\frac{5}{a}, 0)$ , and $y=3x+b$ intersects the $x$ -axis at $(-\frac{b}{3}, 0)$ . We are given that the 2 graphs intersect the x-axis at the same point, so $-\frac{5}{a}=-\frac{3}{b}$ , so $ab=15$ .

The only possible pairs $(a,b)$ then are $(a,b) = (1,15), (3,5), (5,3), (15, 1)$ . These pairs give respective $x$ -values of $-5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}$ which have a sum of $\boxed{\textbf{(E)} \: -8}$ .

2014 AMC 10A (Problems • Answer Key • Resources)
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2014 AMC 10A Problems/Problem 21

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