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? 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
eq 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
eq 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) step 3 is incorrect. the lines do not need to have a point of intersection since they are not parallel. 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). misha did all steps correctly.

Explanation:

Step1: Analyze Step 1

The initial assumption for proof - by - contradiction is valid.

Step2: Analyze Step 2

For two distinct lines \(y = mx + b\) and \(y=mx + c\) with the same slope \(m\), \(b
eq c\) is correct.

Step3: Analyze Step 3

If two lines are not parallel in a two - dimensional plane, they must intersect. This step is correct.

Step4: Analyze Step 4

Setting the two equations equal to find the intersection point is a correct approach.

Step5: Analyze Step 5

When we set \(mx + b=mx + c\), subtracting \(mx\) from both sides gives \(b = c\), which contradicts the condition \(b
eq c\) from Step 2. This step is correct.

Step6: Analyze Step 6

Based on the contradiction, the original assumption is wrong, and the conclusion that two distinct lines with the same slope are parallel is correct.

Answer:

Misha did all steps correctly.