AMC 10 · 2013 · #14

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
polynomial-factoringcoordinate-geometry convert-to-algebracasework ↑ Prerequisites: polynomial-factoring
📏 Medium solution 💡 2 insights
Problem
A made-up operation is defined by a ♣ b = a²b - ab². Find every point (x, y) where using x first gives the same value as using y first, that is, where x ♣ y = y ♣ x, and describe what that set of points looks like.

Pick an answer.

(A)
a finite set of points
(B)
one line
(C)
two parallel lines
(D)
two intersecting lines
(E)
three lines

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

The strange ♣ symbol is just shorthand for an algebra expression, so Tool #13 (Convert to Algebra) turns x ♣ y = y ♣ x into a plain equation you can simplify and factor. Once it is factored into a product that equals zero, Tool #7 (Identify Subproblems) splits it into separate small equations, one for each factor. Tool #1 (Draw a Diagram) then reads each of those equations as a line in the plane, so counting the lines answers the question.

1STEP 1

Write both sides with the rule

Swap each ♣ for the rule: x ♣ y = x²y - xy², while y ♣ x = y²x - yx² = xy² - x²y — the same expression with its sign flipped.

x ♣ y = x²y - xy², y ♣ x = y²x - yx² = xy² - x²y
2STEP 2

Set equal and factor

Set them equal and move everything left: 2x²y - 2xy² = 0. Pull out the common factor 2xy to get 2xy(x - y) = 0.

x²y - xy² = xy² - x²y → 2x²y - 2xy² = 0 → 2xy(x - y) = 0
3STEP 3

Split into three cases

A product is zero only if a factor is zero, and 2 never is — so x = 0, y = 0, or x = y.

2xy(x - y) = 0 ⇔ x = 0 or y = 0 or x = y
4STEP 4

Read each case as a line

Those are the y-axis, the x-axis, and the 45° line y = x — a union of three lines, choice (E).

x = 0, y = 0, y = x → (E)
Answer
three lines
Test a point off all three lines, say (2, 1): x ♣ y = 4·1 - 2·1 = 2 while y ♣ x = 1·2 - 1·4 = -2, so they differ, as expected since (2,1) lies on none of x=0, y=0, y=x. Now test (3, 3) on the line y=x: both sides give 9·3 - 3·9 = 0, so they match. The three lines all pass through the origin, meaning they intersect rather than stay parallel, which rules out (C) and (D); having three of them rules out (A) and (B), leaving (E).
💡Key takeaway

Turn the weird symbol into plain algebra, get one side to zero and factor; each factor that can be zero is its own line, and here there are three of them.

  • Write both sides with the rule
  • Set equal and factor
  • Split into three cases
  • Read each case as a line

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