2016 AMC 12A Problem 7

Below is the professionally curated solution for Problem 7 of the 2016 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12A solutions, or check the answer key.

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Concepts:factoring difference of squares

Difficulty rating: 1410

Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)?$

two parallel lines

two intersecting lines

three lines that all pass through a common point

three lines that do not all pass through a common point

a line and a parabola

Solution:

Moving all terms to one side gives $(x^2-y^2)(x+y+1)=0,$ which factors as $(x-y)(x+y)(x+y+1)=0.$ The graph is therefore the union of the lines $x=y,$ $x=-y,$ and $x+y+1=0.$

The first two lines intersect at the origin, but the third line $x+y=-1$ is parallel to $x+y=0$ and does not pass through the origin. So the graph consists of three lines that do not all pass through a common point.

Thus, the correct answer is D.

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