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 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.
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The incorrect step in Misha's proof is:
Step 2 is incorrect. In order to be distinct lines, it must be true that b is equal to c.
This statement is incorrect because for two distinct lines to exist, their y-intercepts (b and c) must be different (b ≠ c). Misha correctly stated that to be distinct lines, it must be true that b≠c, but the faulty conclusion led her through an erroneous logical path that contradicts her initial assumptions about the lines being distinct.
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