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# Systems of equations: trolls, tolls (1 of 2)

CCSS Math: 8.EE.C.8, 8.EE.C.8b, HSA.REI.C.6, HSA.REI.D.11

## Video transcript

You are traveling in some type
of a strange fantasy land. And you're trying
to get to the castle up here to save the
princess or the prince or whomever you're
trying to save. But to get there, you
have to cross this river. You can't swim across it. It's a very rough river. So you have to
cross this bridge. And so as you approach the
bridge, this troll shows up. That's the troll. And he says, well, I'm
a reasonable troll. You just have to pay $5. And when you look a
little bit more carefully, you see that there
actually was a sign there that says $5 toll
to cross the bridge. Now, unfortunately
for you, you do not have any money in your pocket. And so the troll says,
well, you can't cross. But you say, I need to really,
really get to that castle. And so the troll says, well,
I'll take some pity on you. Instead of paying the $5,
I will give you a riddle. And the riddle is this. And now, I'm speaking
as the troll. I am a rich troll
because I get to charge from everyone who
crosses the bridge. And actually, I only
accept $5 or $10 bills. It's a bit of a
riddle why they accept American currency in
this fantasy land. But let's just take
that as a given for now. So I only take $5 or $10 bills. I'm being the troll. Obviously, if you give me a
$10, I'll give you $5 back. And I know, because I count
my money on a daily basis. I like to save my
money as the troll. I know that I have a
total of 900 bills. So let me write that down. I have a total of 900 bills, a
total of 900 $5 and $10 bills. And he says, because
I'm very sympathetic, I'll give you another
piece of information. He says, if you were to
add up the value of all of my money, which is all in
$5 and $10 bills that I have, I, speaking as the troll about
dollars bills, is $5,500. This is a rich troll. And so the riddle
is-- exactly, exactly. And if you give the wrong
answer and if you're not able to solve it
in 10 minutes, he's just going to push
you into the river or do something horrible to you. He says, exactly how many 5's
and 10's do I, the troll, have? So the first thing I'm going
to have you think about is, is this even a
solvable problem? Because if it's not
a solvable problem, you should probably
run as fast as you can in the other direction. So now, I will tell you, yes,
it is a solvable problem. And let's start thinking about
it a little bit algebraically. And to do that, let's
just set some variables. And I will set the
variables to be what we're really
trying to solve for. We're trying to solve for the
number of $5 bills we have and the number of $10
bills that we have. So let's just define
some variables. I'll say f for 5. Let's let f equal the number
of $5 bills that we have. And I'll use the same idea. Let's let t is equal to the
number $10 bills that we have. Now, given this
information, and now I'm not sure if I'm
speaking as-- well, let's say I'm still
speaking as the troll. I'm a very sympathetic
troll, and I'm going to give you hints. Given this information and
setting these variables in this way, can I represent
the clues in the riddle mathematically? So let's focus on
the first clue. Can I represent this clue that
the total of 900 $5 and $10 bills, or can I represent
that mathematically, that I have a total of
900 $5 and $10 bills? Well, what's going to
be our total of bills? It's going to be the number of
5's that we have, which is f. The number of 5's
that we have is f. And then the number of
10s that we have is t. The total number of 5's
plus the total number of 10s, that's our
total number of bills. So that's going to
be equal to 900. So this statement, this
first clue in our riddle, can be written
mathematically like this if we defined the
variables like that. And I just said f for 5,
because f for 5 in t for 10. Now, let's look at
the second clue. Can we represent this
one mathematically given these variable definitions
that we created? Well, let's think separately
about the value of the $5 bills and the value of the $10 bills. What's the value of
all of the $5 bills? Well, each $5 bill is worth $5. So it's going to be 5 times
the number of $5 bills that we have. So if I have one $5
bill, it will be $5. If I have 100 $5 bills,
then it's going to be $500. How ever many $5 bills,
I just multiply it by 5. That's the value
of the $5 bills. Let me write that down. Value of the $5 bills. Now, same logic. What's the value
of the $10 bills? Well, the value of
the $10 bills is just going to be 10 times
however many bills I have-- value of the $10 bills. So what's going to be the
total value of my bills? Well, it's going to be
the value of the $5 bills plus the value of the $10 bills? And he tells me what
that total value is. It's $5,500. So if I add these
two things, they're going to add up to be $5,500. So this second statement we
can represent mathematically with this second
equation right over here. And what we essentially
have right over here, we have two equations. Each of them have two unknowns. And just using one
of these equations, we can't really figure
out what f and t are. You can pick a bunch of
different combinations that add up to 900 here. You could pick a bunch of
different combinations, where if you work out all the
math, you get $5,500. So independently,
these equations, you don't know what f and t are. But what we will see over
the next several videos is that if you use both of
this information, if you say that there's
an f and a t that has to satisfy both
of these equations, then you can find a solution. And this is called a
system of equations. Let me write that down--
system of equations.