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
.

1 answer

The incorrect step in Misha's proof is Step 3.

Misha assumes that since lines \( l \) and \( n \) are not parallel, they must intersect at some point. However, if two lines are not parallel, it can imply that they intersect; but if they share the same slope and are distinct (as stated), they cannot intersect at any point, because parallel lines never intersect by definition.

Thus, the statement in Step 3 is flawed, because distinct lines with the same slope cannot intersect regardless of any conditions.

Therefore, the correct response is: Step 3 is incorrect. The lines do not need to have a point of intersection since they are not parallel.