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?

Question

1. Assume that distinct line l and n have the same slope, m, but are not parallel
2. 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
3. Since they assumed to not be parallel, they must have a point of intersection
4. Set the equations for l 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

a. Step 2 is incorrect. In order to be distinct lines, it must be true that b = c.
b. Step 5 is incorrect. Both equations need to be solved for x first, then set equal to each other in order to directly solve for the x-intercept
c. Step 3 is incorrect. The lines do not need to have a point of intersection since they are not parallel
d. Misha did all steps correctly

Answer 1

The step that Misha did incorrectly is:

c. Step 3 is incorrect. The lines do not need to have a point of intersection since they are not parallel.

For two distinct lines to not be parallel and have the same slope, they would have to meet at exactly one point, which would mean they have a point of intersection. However, if they are distinct with the same slope, they will actually never intersect, thus actually making them parallel by definition. Therefore, Misha's assumption in step 3 that they must have a point of intersection is incorrect.