AMC 10 · 2016 · #7

Grade 9 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
polynomial-factoringdifference-of-squareszero-product-propertyslope-intercept polynomial-factoringcasework ↑ Prerequisites: difference-of-squareszero-product-property
📏 Medium solution 💡 3 insights
Problem
An equation's graph is every point that makes it true. Identify the shape.

Pick an answer.

(A)
two parallel lines
(B)
two intersecting lines
(C)
three lines that all pass through a common point
(D)
three lines that do not all pass through a common point
(E)
a line and a parabola
How to solve
Strategy Organize Information in More Ways

The tempting move is to cancel the shared factor (x+y+1) and be left with x² = y². That move is not legal here, because (x+y+1) is allowed to be zero, and every point where it is zero already satisfies the equation no matter what x² and y² are. So instead of dividing, reorganize: move everything to one side and factor until the equation reads (product) = 0. A product is zero exactly when one of its factors is zero, which splits the problem into separate, complete cases with nothing thrown away. Each case turns out to be a line, and a quick sketch of the lines settles which description fits.

1STEP 1

Move everything to one side

Everything moves to one side.

x²(x+y+1)-y²(x+y+1)=0 ⟹ (x+y+1)(x²-y²)=0
2STEP 2

Factor the difference of squares

A difference of squares factors further.

(x+y+1)(x²-y²)=(x+y+1)(x+y)(x-y)=0
3STEP 3

Split into three cases

That gives three separate cases.

x+y+1=0 or x+y=0 or x-y=0
4STEP 4

Read off the three lines

Each case is a straight line.

y=x, y=-x, y=-x-1
5STEP 5

Check for a shared point and choose

Two of them are parallel, choice (D).

y=x ∩ y=-x → (0,0); y=x ∩ y=-x-1 → (-1/2,-1/2); y=-x ∥ y=-x-1
Answer
three lines that do not all pass through a common point
Test points against the original equation. On y = x take (2, 2): both sides are 4 * 5 = 20. On y = -x take (3, -3): both sides are 9 * 1 = 9. On y = -x-1 take (2, -3): x+y+1 = 0, so both sides are 0, yet x² = 4 and y² = 9 are different — confirming that this line comes only from the factor that was never cancelled. A point off all three lines, such as (1, 0), gives 1 * 2 = 2 on the left and 0 on the right, so it is correctly excluded. The count also fits: the equation has total degree 3, and a degree-3 curve that splits completely splits into exactly three lines, which is what was found.
💡Key takeaway

Never cancel a factor that could be zero — move everything to one side, factor, and let each factor equal zero be its own branch of the answer.

  • Move everything to one side
  • Factor the difference of squares
  • Split into three cases
  • Read off the three lines
  • Check for a shared point and choose

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