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## Heart of algebra

# Linear function word problems — Harder example

## Video transcript

- [Narrator] Minli's house is located 1.4 miles from her school. When she walks home from school, it takes her an average of 24 minutes. Assuming that Minli
walks at a constant rate, constant rate, we can figure that out, because we know how far she walks in a certain amount of time. Which of the following
functions best models Minli's distance from home, D, in miles. So we want distance in miles. If she has walked a total of T minutes. So T is going to be in minutes, so all our units are gonna
be minutes and miles, and that's good because
they gave us things in terms of minutes and miles. If this was in seconds or hours, we would have to do some conversion. So let's just think about her rate, her constant rate, that
they're talking about. The rate that Minli walks. Well she covers 1.4 miles, she covers 1.4 miles, in 24 minutes, 24 minutes. Well this isn't a pretty fraction with the decimal in the numerator so let's multiply the numerator
and the denominator by 10 so that we get rid of this decimal. So that's going to be equal to
14 over 240 miles per minute. And then we can further simplify that. The numerator and the denominator
are both divisible by two, so this is going to be, let's see, 14 divided by two is seven. 240 divided by two, over
120, miles per minute. Miles per minute. So we were able to figure
out Minli's constant rate. Now we need to figure out D. We need to figure out D. We have to be very careful. D is Minli's distance
from home, from home. She's leaving from school and
it's her distance from home. Remember her home is 1.4
miles from her school. So there's a bunch of ways
that we can tackle it, but maybe the easiest one is, well, what's her distance if you wanted to say distance from school. Distance from school. Distance from school. That's just going to be her rate times T, times how much time has passed by. As when time is zero, she's
going to be at her school. As time increases in
minutes, she's going to get further and further away from her school. But her distance from home, her distance from home,
is going to be 1.4 miles minus the distance from school. So, let me just write D, 'cause this is what we care about, D, which is her distance,
in miles, from home. Distance in miles from home. That's going to be 1.4,
minus distance from school. Minus distance from school. And if that doesn't make
sense, just think about it. If this is her home right over here, I'll write H for home. If this is her school
right here, S for school, we know that this distance
right over here is 1.4 miles. Now she has walked... If she has walked, say... I don't know, let me do
this in another color. If she has walked .4 miles, if this is 0.4 miles right over here, then her distance from home is gonna be 1.4 miles minus that. It's going to be, it's going to be this
distance right over here. So distance from home is going to be 1.4 minus the distance from school. And what's that going to be? Well that's just going to be 1.4 minus the rate times the time. What's the rate? Seven 120ths miles per minute. We got the units right, and so this is D, is going to be equal to
1.4 minus seven over 120 T. And if we look at the choices, well that's going to be
this first choice over here. And it's fun to look at the other choices and to think about well how could we have ruled them out fairly quickly? Well this one has 1.4, the distances between the
two places, minus 24T. Well this isn't the rate right over here, that's how long it takes her
to walk, it's not the rate, so you can rule that one out. This one is 1.4 minus the
reciprocal of the rate, so that's a strange answer. And this one, this one makes it look like she's getting further
and further from home as time gets bigger. Notice this one right over here, it does correctly say at time zero she's going to be 1.4 miles from her home, which is accurate because at time zero she's going to be at her school. But then after one
minute, after two minutes, after three minutes, she's going to get, based on this model, further
and further away from her home. So this would be a case where she's walking away from home from her school, so you would rule that one out as well.