Question

Misha writes the following proof that two distinct lines, I and n, with the same slope, n, are parallel. She uses a proof by contradiction. Which step did she do incorrectly? 1. Assume that distinct linesI and n have the same slope, m, but are not parallel. 2. Let have the equation y=mx+b and n have the equation y=mx+c. In order to be distinct lines, it must be true that b!= c. 3. Since they are assumed to not be parallel, they must have a point of intersection. 4. Set the equations for and n equal to each other and solve to find the x-coordinate of the point of intersection 5. Setting mx+b equal to mx+c results in b=c , which contradicts the condition that b!= c. 6. Therefore the assumption that two distinct lines with the same slope are not parallel is incorrect. It must be true that two distinct lines with the same slope are parallel. (1 point)

Answer 1

Step 2. She assumes both lines can be written as y = mx + b and y = mx + c. That representation requires a finite slope (non‑vertical lines). If the common "slope" is undefined (vertical lines), the equations are x = k and x = l, not y = mx + b, and the argument as written doesn't cover that case.

Fix: split into cases. If the common slope m is finite the given algebraic contradiction is valid and shows b = c, so the lines coincide (contradiction), hence they must be parallel. If the common slope is undefined both lines are vertical (x = k, x = l) and distinct vertical lines are parallel.