AMC 10 · 2018 · #3

Grade 8 rate-ratio
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Point as the Answer
slope-interceptcoordinate-geometrylinear-equations-two-var dimensional-analysiscoordinate-geometry ↑ Prerequisites: slope-interceptcoordinate-geometry
📏 Short solution 💡 2 insights
Problem
Two lines cross each other at the point (40,30). One has slope 2 and the other has slope 6. Each line also crosses the x-axis at its own point. Find how far apart those two crossing points on the x-axis are.

Pick an answer.

(A)
5
(B)
10
(C)
20
(D)
25
(E)
50
How to solve
Strategy Analyze the Units

Tool #8 (Analyze the Units) leads, because slope is a rate: vertical change per unit of horizontal change. Read that way, the problem stops being about two lines and becomes one question asked twice — from the height 30, how much horizontal room does each line need to get down to the axis? A rate answers exactly that kind of question, and it answers it with a single division instead of a full equation. Tool #1 (Draw a Diagram) keeps the signs honest: a sketch of the two lines falling leftward from (40,30) shows that both intercepts sit to the left of 40, and that the steeper line lands closer, so the final subtraction has a predictable direction. Tool #13 (Convert to Algebra) is held in reserve as the check — writing each line in point-slope form and setting y=0 reaches the same two numbers by a completely separate route.

1STEP 1

Each line is already pinned down

A point and a slope fix a line completely.

(40,30) lies on both lines, with m₁=2 and m₂=6; an x-intercept is a point (x,0) on the line.
2STEP 2

Turn the slope into a horizontal step

Turn the slope into a horizontal step.

m=(0-30)/(x-40) ⟹ x-40=-30/m ⟹ x=40-30/m
3STEP 3

Run the rate for each slope

Apply the same rule to each slope.

m=2: x=40-30/2=40-15=25 m=6: x=40-30/6=40-5=35
4STEP 4

Subtract the two intercepts

The gap between the intercepts is 10.

|35-25|=10 → (B) In general, (40-30/6)-(40-30/2)=30(1/2-1/6)=30·1/3=10
Answer
10
Three checks agree. First, direction: the steeper line should reach the axis sooner, and it does — its intercept 35 is closer to x=40 than the gentle line's 25. Second, plug back in: the point (25,0) and the point (40,30) give slope (30-0)/(40-25)=30/15=2, and (35,0) with (40,30) gives (30-0)/(40-35)=30/5=6, so both intercepts really do lie on the intended lines. Third, the wrong choices are all recognizable slips: (A) 5 is the run of the steep line alone, stopping halfway through the work; (C) 20 is 15+5, adding the two runs as though the lines fell in opposite directions instead of the same direction; (D) 25 is one intercept's coordinate reported as if it were the distance; (E) 50 is the size of the y-intercept of the slope-2 line, y=2x-50, which answers a different question. The correct value 10 is the only one produced by subtracting two runs measured from the same starting point.
💡Key takeaway

Slope tells you what a fall costs in sideways travel: dropping 30 takes a run of 15 at slope 2 but only 5 at slope 6, so the two landing spots end up 10 apart.

  • Each line is already pinned down
  • Turn the slope into a horizontal step
  • Run the rate for each slope
  • Subtract the two intercepts

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