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 when $y=0$ , the $x$ values of the equations should be equal by the problem statement. We have that $0 = ax + 5 \implies x = -\dfrac{5}{a}$ $0 = 3x+b \implies x= -\dfrac{b}{3}$ Which means that $-\dfrac{5}{a} = -\dfrac{b}{3} \implies ab = 15$ 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 $\boxed{\textbf{(E)} \: -8}$ .

--happiface.

2014 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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2014 AMC 10A Problems/Problem 21

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