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?

Question

1. Assume that distinct lines l and n have the same slope, m , but are not parallel.
2. 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.
3. Since they are assumed to not be parallel, they must have a point of intersection.
4. Set the equations for l and n equal to each other and solve to find the x -coordinate of the point of intersection.
5. Setting mx+b equal to mx+c results in b=c , which contradicts the condition that b≠c.
6. 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.

A. Step 3 is incorrect. The lines do not need to have a point of intersection since they are not parallel
B. 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.
C. Misha did all steps correctly.
D. Step 2 is incorrect. In order to be distinct lines, it must be true that b=c .

Answer 1

The correct evaluation of Misha's proof is:

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

Here's why: In her assumption, Misha states that the lines l and n are not parallel. If two lines are not parallel, they either intersect at one point or are the same line. In the context of her proof, since she is assuming they are distinct lines with the same slope (thus, they can't be the same line), she should conclude that they cannot have a point of intersection if they are not parallel and have the same slope. Therefore, step 3 does not hold true under the assumption that the lines are not parallel, which is where the contradiction arises.

Steps 2, 5, and the overall logical flow of her proof are structured correctly, leading to the false conclusion of distinct lines intersecting when assumed not to be parallel, but sharing the same slope.