Misha writes the following proof that two distinct lines, I and n, with the same slope, m, are parallel. She uses a proof

Question

Misha writes the following proof that two distinct lines, I and n, with the same slope, m, are parallel. She uses a proof
by contradiction. Which step did she do incorrectly?
1. Assume that distinct lines and n have the same slope, m, but are not parallel.
2. Let I have the equation y = mx + b and n have the equation ya
= 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 I and n equal to each other and solve to find the x-coordinate of the point of intersecti-
5. Setting mz+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
that two distinct lines with the same slope are parallel.
(1 point)
 
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.

Misha did all steps correctly

Step 2 is incorrect. In order to be distinct lines, it must be true that b = c

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

GPT-4o mini · Answer

The incorrect step in Misha's proof is:

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

In her proof by contradiction, Misha assumes that the two lines are distinct and not parallel. For two lines to not be parallel, they must intersect at some point. Therefore, step 3 is flawed in its reasoning because if the lines are not parallel, they must intersect.

The correct reasoning would be that since the lines are not parallel (as per her assumption), they should intersect at some point, which leads to a contradiction due to the conclusion reached in the final step about the slopes (which were assumed to be the same). So, the correct issue lies with her claim that if the lines are not parallel, they do not need to intersect, which is false.