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Problem 261

Algebra Difficulty 2.0 Find the answer CEMC Cayley

For what value of $k$ is the line through the points $(3,2k+1)$ and $(8,4k-5)$ parallel to the $x$ -axis?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solutions — 2

Solution 1

A line segment joining two points is parallel to the $x$ -axis exactly when the $y$ -coordinates of the two points are equal. Here, this means that $2k+1=4k-5$ and so $6=2k$ or $k=3$ . (We can check that when $k=3$ , the coordinates of the points are $(3,7)$ and $(8,7)$ .)

Solution 2

A line segment joining two points is parallel to the $x$ -axis exactly when the $y$ -coordinates of the two points are equal. Here, this means that $2k+1 = 4k-5$ and so $6 = 2k$ or $k = 3$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.