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

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.

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.

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.

Misha did all steps correctly.

Answer 1

The incorrect step in Misha's proof is:

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

In Misha's proof, she assumes that if the lines are not parallel, they must intersect at a point. However, if two lines are distinct and have the same slope, they cannot intersect, which is the definition of parallel lines. Therefore, the step asserting that the two lines must have a point of intersection is incorrect.

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.

Answer 2

Which of the following graphed lines is parallel to y=34x+3?(1 point)
Responses

A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.
Image with alt text: A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.

A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.
Image with alt text: A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.

A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.
Image with alt text: A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.

A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.