Misha writes the following proof that two distinct lines, l and n , with the same slope, m , are parallel. She uses a proof by contradiction. Which step did she do incorrectly? Assume that distinct lines l and n have the same slope, m , but are not parallel. Let l 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 . Since they are assumed to not be parallel, they must have a point of intersection. Set the equations for l and n equal to each other and solve to find the x -coordinate of the point of intersection. Setting mx+b equal to mx+c results in b=c , which contradicts the condition that b≠c . 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.

Question

GPT-4o mini · Answer

The error in Misha's proof lies in the assumption that distinct lines \(l\) and \(n\) with the same slope \(m\) can intersect. The key fact to recognize is that if two distinct lines have the same slope, they must be parallel.

Here's the flaw in reasoning in more detail:

Misha starts with the equations \(y = mx + b\) (for line \(l\)) and \(y = mx + c\) (for line \(n\)), where \(b \neq c\) ensures the lines are distinct.
She states that since the lines are distinct, they must intersect at a point. However, this statement is where the assumption begins to fail. If line \(l\) and line \(n\) have the same slope, they will never intersect because parallel lines do not meet.
Consequently, when she sets \(mx + b = mx + c\) and derives \(b = c\), she is arriving at a contradiction based on incorrect premises. The contradiction arises not because her calculations are wrong, but because the initial assumption that two distinct lines with the same slope can intersect is false.

Therefore, Misha's mistake is in the assumption that distinct lines with the same slope can intersect. This is incorrect because parallel lines, by definition, do not intersect. Thus, she should conclude that if the lines are distinct and have the same slope, they must be parallel.