Question

Misha writes the following proof that two distinct lines, I and n, with the same slope, n, are parallel. She uses a proof by contradiction. Which step did she do incorrectly? 1. Assume that distinct linesI and n have the same slope, m, but are not parallel. 2. Let 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 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. (1 point)
A. Step 2 is incorrect. In order to be distinct lines, it must be true that b = c.
B. Sten 5 is incorrect. Both equations need to be solved for r first, then set equal to each other in order to directly solve for the x-intercept
C. Step 3 is incorrect. The lines do not need to have a point of intersection since they are
not parallel.
D. Misha did al steps correctly

Answer 1

D. Misha did all steps correctly.

Her contradiction is valid: if two distinct lines have the same slope m, write them y = mx + b and y = mx + c with b ≠ c. Assuming they are not parallel forces them to intersect, so mx + b = mx + c at the intersection, which implies b = c — contradicting b ≠ c. Hence the assumption is false and distinct lines with the same slope are parallel.