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 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.

Misha did all steps correctly.

Answer 1

Misha did all steps correctly.

Reason: Writing l: y = mx + b and n: y = mx + c with b ≠ c expresses that the lines are distinct. Assuming they are not parallel implies they intersect, so equating mx + b = mx + c is valid; subtracting mx gives b = c, contradicting b ≠ c. Hence the assumption is false and the distinct lines with the same slope must be parallel.