Question

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 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. 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 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.

Answer 1

The key issue in Misha's proof occurs in Step 2.

In this step, she states that for two distinct lines to exist, it must be true that \( b \neq c \). This part is correct: distinct lines cannot have the same y-intercept (intercept \( b \) in line \( l \) and \( c \) in line \( n \)) if they are to be distinct. However, she incorrectly assumes they can still intersect while being composed of the same slope. According to the properties of lines, if two lines have the same slope but different y-intercepts, they cannot intersect, meaning they are indeed parallel.

Thus, the incorrect assumption about the existence of a point of intersection for these distinct, same-slope lines leads to a contradiction, and that contradiction is indeed valid for her conclusion. Nonetheless, the error does lie in Step 2 where she misrepresents the distinction for such lines.

Therefore, the correct identification of the error in Misha's proof is:

Step 2 is incorrect. In order to be distinct lines, it must be true that \( b \neq c \).