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 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 . Step 2 is incorrect. In order to be distinct lines, it must be true that b is equal to c. Misha did all steps correctly.

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The incorrect step in Misha's proof is Step 3. She assumes that if two distinct lines are not parallel, then they must have a point of intersection. However, this is not necessarily true if the lines are distinct and have the same slope. In fact, lines that are not parallel cannot intersect and thus cannot have a point of intersection, which contradicts the premise that they are not parallel. Therefore, her reasoning in Step 3 is flawed.

So, the correct response would be: 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 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 . Step 2 is incorrect. In order to be distinct lines, it must be true that b is equal to c. Misha did all steps correctly.

1 answer

The incorrect step in Misha's proof is Step 3. She assumes that if two distinct lines are not parallel, then they must have a point of intersection. However, this is not necessarily true if the lines are distinct and have the same slope. In fact, lines that are not parallel cannot intersect and thus cannot have a point of intersection, which contradicts the premise that they are not parallel. Therefore, her reasoning in Step 3 is flawed.

So, the correct response would be: Step 3 is incorrect. The lines do not need to have a point of intersection since they are not parallel.

GPT-4o mini · Answer

The incorrect step in Misha's proof is **Step 3**. She assumes that if two distinct lines are not parallel, then they must have a point of intersection. However, this is not necessarily true if the lines are distinct and have the same slope. In fact, lines that are not parallel cannot intersect and thus cannot have a point of intersection, which contradicts the premise that they are not parallel. Therefore, her reasoning in Step 3 is flawed. So, the correct response would be: **Step 3 is incorrect. The lines do not need to have a point of intersection since they are not parallel.**