AMC 10 · 2014 · #21

Grade 8 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

linear-equations-two-varslope-interceptfactors convert-to-algebrasystematic-enumeration ↑ Prerequisites: linear-equations-two-var

📏 Medium solution 💡 2 insights

Problem

Two lines, y=ax+5 and y=3x+b with positive integers a and b, cross the x-axis at the same point. Find the sum of every possible x-coordinate where that shared crossing can happen.

Pick an answer.

(A)

${-20}$

(B)

${-18}$

(C)

${-15}$

(D)

${-12}$

(E)

${-8}$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The phrase "cross the x-axis at the same point" is a geometry sentence, so Tool #4 (Introduce a Variable) converts it into algebra: write each line's x-intercept in terms of a and b, then set them equal. That collapses the whole condition into one equation, ab=15. Tool #2 (Make a Systematic List) then finds every positive-integer pair (a,b) with ab=15, guaranteeing we catch all possible crossing points. Tool #7 (Identify Subproblems) keeps the work in clean stages — intercept, factor pairs, coordinates, sum — instead of one tangled calculation.

1STEP 1

Find each line's x-axis crossing

On the x-axis y=0, so 0=ax+5 gives x=-5/a and 0=3x+b gives x=-b/3.

0=ax+5→ x=-5/a; 0=3x+b→ x=-b/3

2STEP 2

Set the two crossings equal

The crossings are shared, so -5/a=-b/3; cross-multiplying collapses the whole condition to ab=15.

-5/a=-b/3→ 5/a=b/3→ ab=15

3STEP 3

List the positive factor pairs of 15

The positive divisors 1,3,5,15 give exactly four pairs: (1,15),(3,5),(5,3),(15,1).

ab=15→ (a,b)∈{(1,15),(3,5),(5,3),(15,1)}

4STEP 4

Turn each pair into its x-coordinate

Putting each a into x=-5/a gives four distinct crossings: -5, -5/3, -1, -1/3.

x=-5/a: -5, -5/3, -1, -1/3

5STEP 5

Add the x-coordinates

Grouping wholes and thirds, -5-1=-6 and -5/3-1/3=-2, so the total is -8, choice (E).

-5-5/3-1-1/3=(-5-1)+(-5/3-1/3)=-6-2=-8→(E)

Answer

{-8}

Each crossing sits at a negative x between -5 and -1/3, so four of them must add to something negative and no smaller than 4×(-5)=-20; -8 lands comfortably inside that range. A common slip is forgetting the reversed pairs (5,3) and (15,1), but the four distinct values -5,-5/3,-1,-1/3 are exactly the factor pairs of 15, and they sum to -8, matching (E).

💡Key takeaway

Turn "same x-axis crossing" into the equation ab=15, list its factor pairs, then add up each crossing point.

Find each line's x-axis crossing
Set the two crossings equal
List the positive factor pairs of 15
Turn each pair into its x-coordinate
Add the x-coordinates

Find each line's x-axis crossing

Set the two crossings equal

List the positive factor pairs of 15

Turn each pair into its x-coordinate

Add the x-coordinates

More like this