2014 AMC 10A Problem 21

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

Set both x-intercepts equal

Positive integer factor pairs of

15

give all possibilities

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.

Problem 21 in other years