AMC 10 · 2003 · #11

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometryslope-interceptlinear-equations-one-var convert-to-algebra ↑ Prerequisites: slope-intercept
📏 Medium solution 💡 2 insights
Problem
Two lines both pass through the point (10,15): one has slope 3 and the other has slope 5. Each line crosses the x-axis at its own x-intercept. Find the distance between those two x-intercepts.

Pick an answer.

(A)
2
(B)
5
(C)
7
(D)
12
(E)
20

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Each line is fully determined by its slope and the shared point (10,15), so tool #4 (Introduce a Variable) says: write the point-slope equation of each line with x and y as the variables, then set y=0 to solve for the unknown x-intercept. Tool #8 (Analyze the Units) gives the same result faster: slope is rise over run, so to drop the 15 units from y=15 down to y=0 each line moves a run of 15/slope backward, telling you exactly how far the intercept sits from x=10. Tool #1 (Draw a Diagram) keeps the picture straight — two lines fanning out from one point down to two nearby spots on the x-axis — so the final subtraction of the two x-values is clearly the distance asked for.

1STEP 1

Write each line's equation

Point-slope form through the shared point (10,15) gives y - 15 = 3(x - 10) and y - 15 = 5(x - 10).

y - 15 = 3(x - 10) and y - 15 = 5(x - 10)
2STEP 2

Find the slope-3 x-intercept

Set y = 0 in the slope-3 line: -15 = 3(x - 10), so x - 10 = -5 and it meets the axis at x = 5.

0 - 15 = 3(x - 10) → x - 10 = -5 → x = 5
3STEP 3

Find the slope-5 x-intercept

Same move on the slope-5 line: -15 = 5(x - 10), so x - 10 = -3 and its intercept is x = 7, nearer to 10.

0 - 15 = 5(x - 10) → x - 10 = -3 → x = 7
4STEP 4

Subtract to get the distance

Both intercepts lie on the x-axis, so the distance is just the gap in x-coordinates: |7 - 5| = 2, choice (A).

|7 - 5| = 2 → (A)
Answer
2
Both intercepts, 5 and 7, sit just to the left of x=10, which makes sense because both lines slope upward and must come back down to the axis. The steeper slope-5 line lands at 7 (closer to 10) and the gentler slope-3 line at 5 (farther), a gap of 2 — small, and matching choice (A). Checking a point: on the slope-3 line at x=5, y = 15 + 3(5-10) = 15 - 15 = 0, and on the slope-5 line at x=7, y = 15 + 5(7-10) = 15 - 15 = 0, so both really do hit the axis.
💡Key takeaway

To find where a line hits the x-axis, set y=0 and solve for x; the distance between two intercepts is just the difference of their x-values.

  • Write each line's equation
  • Find the slope-3 x-intercept
  • Find the slope-5 x-intercept
  • Subtract to get the distance

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