Question

The graphs of $2x+3y-6=0,$ $4x-3y-6=0,$ $x=2,$ and $y=\dfrac23$ intersect in:

$6$ points

$1$ point

$2$ points

no points

an unlimited number of points

Answer: B

Concepts:linear equations common intersection coordinate geometry

Difficulty rating: 1340

Small Hint:

Solve the first two linear equations simultaneously

Big Hint:

Check whether that solution also satisfies each of the last two displayed equations

Solution:

Adding the first two equations gives $6x-12=0,$ so $x=2.$ Substitution gives $3y=2,$ hence $y=\frac{2}{3}.$ This point also lies on the last two given lines. Therefore all four graphs have the single common point $(2,\frac{2}{3}).$

Thus, the correct answer is B.

1955 AMC 12 Problem 46

Answer: B

Difficulty rating: 1340

Small Hint:

Big Hint:

Solution:

Problem 46 in Other Years