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 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 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 .

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The error in Misha's proof is in Step 3. The key step that she incorrectly made is the assumption that two distinct lines with the same slope must have a point of intersection.

In reality, if two lines are distinct and have the same slope, they cannot intersect at any point; this is precisely the definition of parallel lines. Therefore, Misha's claim that since they are assumed to not be parallel, they must intersect, is incorrect.

Consequently, Misha's statement and reasoning lead to a contradiction that does not accurately reflect the relationship between distinct lines with the same slope. 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.

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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 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 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 .

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The error in Misha's proof is in Step 3. The key step that she incorrectly made is the assumption that two distinct lines with the same slope must have a point of intersection.

In reality, if two lines are distinct and have the same slope, they cannot intersect at any point; this is precisely the definition of parallel lines. Therefore, Misha's claim that since they are assumed to not be parallel, they must intersect, is incorrect.

Consequently, Misha's statement and reasoning lead to a contradiction that does not accurately reflect the relationship between distinct lines with the same slope. 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.

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GPT-4o mini · Answer

The error in Misha's proof is in **Step 3**. The key step that she incorrectly made is the assumption that two distinct lines with the same slope must have a point of intersection. In reality, if two lines are distinct and have the same slope, they cannot intersect at any point; this is precisely the definition of parallel lines. Therefore, Misha's claim that since they are assumed to not be parallel, they must intersect, is incorrect. Consequently, Misha's statement and reasoning lead to a contradiction that does not accurately reflect the relationship between distinct lines with the same slope. 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.**