Parallel and Perpendicular Lines Quick Check

Question

2 of 52 of 5 Items
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

Misha did all steps correctly.
Misha did all steps correctly.

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.

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 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.
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Answer 1

The mistake in Misha's proof is in Step 3.

The lines \( l \) and \( n \) with the same slope \( m \) and different y-intercepts \( b \) and \( c \) do not intersect because they are parallel. Therefore, it is incorrect to assume that they must have a point of intersection.

The correct interpretation is that two lines with the same slope are parallel if and only if their y-intercepts are different. Misha's conclusion that they must intersect leads to a faulty contradiction, as parallel lines do not intersect at all.

Thus, the correct response is:

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