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# Modeling with linear equations: snow

Sal uses a linear equation to model the amount of snow on the ground. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

On Monday morning, there
were 12 inches of snow on the ground. The weather warmed up, and
by Tuesday morning, 2 inches had melted. All right, so we'll
have 10 left. 2 more inches melted by
Wednesday morning. This pattern continued
throughout the week until no more snow was left. So they're essentially saying
that we had 12 inches of snow on the ground on Monday and that
every day after that, two inches melted. So after Tuesday, you'd have
10 inches, and after Wednesday, you'd have
eight inches, and that pattern continued. And what they say is create an
equation and a graph to show the relationship between the day
and the amount of snow on the ground. So let's define a variable that
tells us how far away we are from Monday. So let's let x equal
days after Monday. And then let y be equal to
inches of snow on the ground. So, one way to think about it
is, OK, when x is 0, when we're on Monday, when we're 0
days after Monday, we're going to have 12 inches of snow on
the ground, and every day after that, we're going
to lose two inches. So if we're on Tuesday, we're
going to have 2 inches times 1, because Tuesday is one day,
so if x is 1, that means we're on Tuesday. If x is 2, that means we're 2
times 2, we've lost 4 inches, which is what the case
is on Wednesday. So this is our equation for the
relationship between the day and the amount of
snow on the ground. x is the day, how many days
after Monday, and then y is the inches of the snow
left on the ground. We start with 12, and
then every day we lose exactly two inches. Now let's graph this. So I'll make my vertical axis
the y-axis, that's inches on the ground. y is equal to inches
left on the ground. And then the horizontal axis,
that is our x-axis-- let me scroll down a little bit-- this
is days after Monday. And so we have 0 days after
Monday, we have 1, 2, 3, 4, 5, and 6. And then on Monday, which is
exactly 0 days after Monday, that is Monday, we have 12
inches on the ground. So I'll do it up here, so we
have 12 inches on the ground right there. And actually, I could do
a table if you like. Let me draw this. So if we do x and y, this is
the days after Monday, so there's 0, 1, 2, 3, 4, 5, 6. And then on the first day, we
have 12 inches, on Monday, 0 days after Monday. Then we lose two inches
each day. On day 1 we have 10, day
2, 8, 6, 4, 2, 0. So let's plot these points. We already plotted 0, 12
in that blue color. Now let's plot 1, 10. 1, 10 is right about there. It'll be right over there. Then we can plot 2, 8. So 2 days after Monday,
we have 8 inches left on the ground. So this is on Wednesday,
so that's 8 inches. And then on 3 days after Monday,
we have 6 inches on the ground. You can see that a line
is forming here. And then if we go to 4 days
after Monday, we have 4 inches on the ground. So that is 4. And then 5 days after
Monday, we have 2 inches on the ground. And then finally, on the sixth
day, 6 days after Monday-- so what are we at, Sunday now-- we
are going to have no inches on the ground. So that's that right there. And you can see that there's
this line that formed, because this is a linear relationship. It looks a little curvy because
I didn't draw it perfectly, but that is a line. So we've done everything. We've created the equation. We start with 12 inches,
every day after that we lose two inches. And we showed a graph that
depicts the relationship.