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# Linear & nonlinear functions: word problem

Video transcript

Luis and Kate have two video
games they want to play. They plan to spend exactly 45
minutes playing the two games. They want to use an
equation to express the relationship between the
number of minutes they spend playing Super Bologna Man
and the number of minutes they spend playing You
Have to Cut the Wire. Can this relationship
be represented using a linear equation? So let's see if it can. Let's define one variable
for the amount of time, the number of minutes they
spend playing Super Bologna Man. Let's define that as, well
let's just say that's x. So x is equal to time
playing Bologna Man. I'll write Bologna right here. And let's define y as y is
equal to the number of minutes they spend playing You
Have to Cut the Wire. So it's time playing, I'll
just call it Wire for short. So if we have the
minutes they play time playing Bologna and the
time playing Cut Your Wire. If I were to add those two
together, so if I say x plus y. I'll write that plus in a
neutral color, x plus y. What does this need
to be equal to? The time I play Bologna
Man plus the time I play Have to Cut the Wire. Well if I add them
together, they want to spend exactly 45
minutes playing both games. So this is going to be
equal to 45 minutes. Now we have set up
an equation that relates the time playing
Bologna Man and the time playing You Have
to Cut the Wire. But now we have to think about
is this a linear relationship? So one way to think about
it is the real giveaway for a linear relationship
is if you can write it in the traditional
form of a line. So if you can write
it in the y is equal to mx plus b form, where
m is the slope of the line and b is the y-intercept. So let's see if we can do that. Well if we want to
do that here, we could just subtract
x from both sides. You subtract x from
both sides, you get-- so let's
subtract an x over here, let's subtract
an x over here. So negative x plus this and
then subtract an x there. Well, that's going
to cancel, and you're going to be left
with-- and I'm going try to write it in this
form right over here-- y is equal to 45. Let me do that in
the same color, just to make it not be confusing. y is equal to, and I'll
write the negative x first because we have the x term
right over here first. So y is equal to
negative x plus 45. All I did is I switched
these two terms around. But you see here,
it has that form. And you might say
wait what is m here? I see that b is 45. Well if I write
negative x, that's the same thing as
writing negative 1x. So this is definitely a line. I was able to write it in
this form right over here. So can this relationship
be represented using a linear equation? Absolutely, absolutely yes.