Geometry & Figures

Depth · one concept from first principles · step 2 of 4

Swapped-coefficient lines always meet at x = 1

Two lines have their coefficients swapped: y=ax+b and y=bx+a. Their crossing point is in Quadrant IV. Given ab ＞ 0 and a ≠ b, decide which quadrant holds (a, b).
ExpressionsNumber system
Your answer
How to solve
Strategy Convert to Algebra — Split it into two small problems. First find the intersection by setting the two expressions equal -- the swap makes the x-coordinate come out as exactly 1. Then the Quadrant IV condition says something about a+b, and together with ab ＞ 0 it pins down both signs.
1STEP 1

Set the two lines equal

At the intersection the two y-values match.

ax+b=bx+a → (a-b)x=a-b
2STEP 2

Divide by a - b

Since a ≠ b, we may divide by a-b. Then substitute x=1 into either line.

x=1, y=a · 1+b=a+b
3STEP 3

Use Quadrant IV

A point in Quadrant IV has a positive x-coordinate and a negative y-coordinate. Here x=1 ＞ 0 already, so the condition is about y.

a+b ＜ 0
4STEP 4

Pin down the signs

ab ＞ 0 means a and b have the same sign. Both positive is ruled out, because then a+b would be positive. So both are negative.

a ＜ 0, b ＜ 0
Answer
Quadrant III
Example: a=-1, b=-2 (so ab=2 ＞ 0). The lines y=-x-2 and y=-2x-1 meet where -x-2=-2x-1, i.e. x=1, y=-3: Quadrant IV, as required. And (a, b)=(-1, -2) is in Quadrant III.
Takeaway

Swap the slope and intercept of a line, and the two lines always cross above or below x=1 -- a symmetry is a free clue.

  • Set the two lines equal
  • Divide by a - b
  • Use Quadrant IV
  • Pin down the signs
Where next?
Same look, different logicsuggested
Find which side the bisector hits first
Step back and solidifysuggested
A second solution forces the two lines to coincideTwo lines through the origin meeting twice coincide

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