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

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.
(1 point)
Responses

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

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.
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.

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

Misha did all steps correctly.

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 Misha's proof, she assumes two distinct lines \( l \) and \( n \) with the same slope \( m \) and that they are not parallel. However, non-parallel lines by definition either do not intersect (if they are parallel) or they intersect at exactly one point (if they are different lines with different slopes). Since she posits that they have the same slope but are not parallel, this leads to a contradiction because lines that share the same slope must indeed be parallel and therefore cannot intersect. Thus, her assumption about the intersection is incorrect.