Misha writes the following proof that two distinct lines, l 𝑙 and n 𝑛 , with the same slope, m 𝑚 , are parallel. She uses a proof by contradiction. Which step did she do incorrectly? Assume that distinct lines l 𝑙 and n 𝑛 have the same slope, m 𝑚 , but are not parallel. Let l 𝑙 have the equation y=mx+b 𝑦 = 𝑚 𝑥 + 𝑏 and n 𝑛 have the equation y=mx+c 𝑦 = 𝑚 𝑥 + 𝑐 . In order to be distinct lines, it must be true that b≠c 𝑏 ≠ 𝑐 . Since they are assumed to not be parallel, they must have a point of intersection. Set the equations for l 𝑙 and n 𝑛 equal to each other and solve to find the x 𝑥 -coordinate of the point of intersection. Setting mx+b 𝑚 𝑥 + 𝑏 equal to mx+c 𝑚 𝑥 + 𝑐 results in b=c 𝑏 = 𝑐 , which contradicts the condition that b≠c 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 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 𝑥 first, then set equal to each other in order to directly solve for the x 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. Misha did all steps correctly.

asked by hihi

5 months ago

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The critical error in Misha's argument occurs in Step 3. The assumption that two distinct lines with the same slope can intersect is incorrect. By definition, if two lines have the same slope and are distinct (not overlapping), they must be parallel and cannot intersect at any point.

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.

Misha's confusion lies in her assumption that if two lines are distinct with the same slope, they can still intersect, which contradicts the definition of parallel lines.

answered by GPT-4o mini

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Question

Misha writes the following proof that two distinct lines, l 𝑙 and n 𝑛 , with the same slope, m 𝑚 , are parallel. She uses a proof by contradiction. Which step did she do incorrectly? Assume that distinct lines l 𝑙 and n 𝑛 have the same slope, m 𝑚 , but are not parallel. Let l 𝑙 have the equation y=mx+b 𝑦 = 𝑚 𝑥 + 𝑏 and n 𝑛 have the equation y=mx+c 𝑦 = 𝑚 𝑥 + 𝑐 . In order to be distinct lines, it must be true that b≠c 𝑏 ≠ 𝑐 . Since they are assumed to not be parallel, they must have a point of intersection. Set the equations for l 𝑙 and n 𝑛 equal to each other and solve to find the x 𝑥 -coordinate of the point of intersection. Setting mx+b 𝑚 𝑥 + 𝑏 equal to mx+c 𝑚 𝑥 + 𝑐 results in b=c 𝑏 = 𝑐 , which contradicts the condition that b≠c 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 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 𝑥 first, then set equal to each other in order to directly solve for the x 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. Misha did all steps correctly.

1 answer

The critical error in Misha's argument occurs in Step 3. The assumption that two distinct lines with the same slope can intersect is incorrect. By definition, if two lines have the same slope and are distinct (not overlapping), they must be parallel and cannot intersect at any point.

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.

Misha's confusion lies in her assumption that if two lines are distinct with the same slope, they can still intersect, which contradicts the definition of parallel lines.

GPT-4o mini · Answer

The critical error in Misha's argument occurs in Step 3. The assumption that two distinct lines with the same slope can intersect is incorrect. By definition, if two lines have the same slope and are distinct (not overlapping), they must be parallel and cannot intersect at any point.

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.

Misha's confusion lies in her assumption that if two lines are distinct with the same slope, they can still intersect, which contradicts the definition of parallel lines.