A linear function that represents the number of animals adopted from the shelter is compared to a different linear function that represents the hours volunteers work at the shelter each week. Describe the key features of the functions that are
needed to determine if these lines intersect.

Two lines only can intersect if the lines are not parallel.

And given a linear equation:

y = a*x + b

Where a is the slope and b is the y-intercept, the general equation for a parallel line to this one is:

y = a*x + c

with the condition:

b ≠ c

So we need to have the same slope and a different y-intercept.

Then when we compare the linear function that represents the number of animals adopted with the linear function that represents the hours volunteers work at the shelter, you need to look if both equations have a different slope or not.

Let's describe the two cases:

If the lines have the same slope, then you need to see the y-intercept.

If the y-intercept is different, then the lines never do intersect.

If the y-intercept is equal, then both equations describe the same line (as both have the same slope and the same y-intercept).

If the lines have different slope, then we can conclude that the lines are not parallel, and the lines will intersect at some point.

both equations and proportions are better.