Question

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?

$-20$

$-18$

$-15$

$-12$

$-8$

Answer: E

Concepts:linear equation factor systematic listing

Difficulty rating: 1600

Solution:

Note that the lines intersect the $x$ -axis when $y = 0.$ This gives us $0 = ax + 5$ and $0 = 3x + b,$ which when solved gives us $x = -\dfrac{5}{a}$ and $x = -\dfrac{b}{3}.$

Setting these equal to each other, we have $\dfrac{5}{a} = \dfrac{b}{3}$ $ab = 15.$

We know that $a$ and $b$ are positive, which means that the only pairs of values $(a, b)$ that satisfy the above equation are $(1, 15),$ $(3, 5),$ $(5, 3),$ $(15, 1).$

Plugging these values back into the equations gives us $x$ -values of $x = -5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}.$ The sum of all these values is $-8.$

Thus, E is the correct answer.

2014 AMC 10A Problem 21

Answer: E

Difficulty rating: 1600

Solution:

Problem 21 in Other Years