Question

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 which answer is correct and why with explanation

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 correct response is:

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

Explanation:

In Misha's proof, she assumes that two distinct lines \( l \) and \( n \) with the same slope \( m \) are not parallel. For lines to be considered not parallel, they must intersect at some point. However, for lines with the same slope to be distinct, they cannot intersect; they are parallel by definition. Therefore, the assertion that if the lines are not parallel, they must intersect, is inherently flawed.

Since the lines are distinct and have the same slope, the condition that \( b \neq c \) must be upheld along with the conclusion that they cannot intersect (and thus cannot have a point of intersection). Thus, the assumption in step 3 leads to a contradiction and demonstrates that the initial assumption (that the lines are not parallel) is indeed incorrect.

In conclusion, the critical error is in the argument regarding the intersection of the lines; the assumption that distinct lines with the same slope could intersect is false, and this leads to the contradiction Misha attempts to establish.