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

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

## Video transcript

Where we left off, we
were trying our very best to get to the castle
and save whomever we were needing to save. But we had to cross the
bridge and the troll gave us these clues because
we had no money in our pocket. And if we don't
solve his riddle, he's going to push
us into the water. So we are under pressure. And at least we made some
headway in the last video. We were able to represent
his clues mathematically as a system of equations. What I want to do in
this video is think about whether we can solve for
this system of equations. And you'll see that
there are many ways of solving a system
of equations. But this time I want
to do it visually. Because at least in my
mind, it helps really get the intuition of what
these things are saying. So let's draw some
axes over here. Let's draw an f-axis. That's the number of
fives that I have. And let's draw a t-axis. That is the number
of tens I have. And let's say that this
right over here is 500 tens. That is 1,000 tens. And let's say this is-- oh,
sorry, that's 500 fives. That's 1,000 fives. This is 500 tens, And
this is 1,000 tens. So let's think about all of
the combinations of f's and t's that satisfy this
first equation. If we have no tens, then
we're going to have 900 fives. So that looks like
it's right about there. So that's the point
0 tens, 900 fives. But what if went the other way? If we have no fives, we're
going to have 900 tens. So that's going to be the
point 900 tens, 0 fives. So all the combinations of
f's and t's that satisfy this are going to be on this
line right over there. And I'll just draw a dotted line
just because it's easier for me to draw it straight. So that represents all the
f's and t's that satisfy the first constraint. Obviously, there's
a bunch of them, so we don't know
which is the one that is actually what the troll has. But lucky for us, we have
a second constraint-- this one right over here. So let's do the same thing. In this constraint, what
happens if we have no tens? If tens are 0, then we
have 5f is equal to 5,500. Let me do a little
table here, because this is a little bit more involved. So for the second
equation, tens and fives. If I have no tens, I have
5f is equal to 5,500, f will be 1,100. I must have 1,100 fives. If I have no fives,
then this is 0, and I have 10t is
equal to 5,500, that means I have 550 tens. So let's plot both
at those point. t equals 0, f is 11. That's right about there. So that is 0. 1,100 is on the line that
represents this equation. And that when f is 0, t is 550. So let's see, this is about--
this would be 6, 7, 8, 9, so 550 is going to
be right over here. So that is the
point 550 comma 0. And all of these
points-- let me try to draw a straight line again. I could do a better
job than that. So all of these points
are the points-- let me try one more time. We want to get this right. We don't want to get pushed
into the water by the troll. So there you go. That looks pretty good. So every point on this blue line
represents an ft combination that satisfies the
second constraint. So what is an f and t, or
number of fives and number of tens that satisfy
both constraints? Well, it would be
a point that is sitting on both of the lines. And what is a point that is
sitting on both of the lines? Well, that's where
they intersect. This point right over here
is clearly on the blue line and is clearly on
the yellow line. And what we can do is, if
we drew this graph really, really precisely, we could
see how many fives that is and how many tens that is. And if you look at it, if
you look at very precisely, and actually I encourage you
to graph it very precisely and come up with how many fives
and how many tens that is. Well, when we do it right over
here, I'm going to eyeball it. If we look at it
right over here, it looks like we
have about 700 fives, and it looks like we
have about 200 tens. And this is based on
my really rough graph. But let's see if that worked. 700 plus 200 is equal to 900. And if I have 700 fives--
let me write this down. 5 times 700 is going to be
the value of the fives, which is $3,500. And then 10 plus 10 times
200, which is $2,000, $2,000 is the value of the 10s. And if you add up
the two values, you indeed get to $ 5,500 So this looks right. And so we can tell
the troll-- Troll! I know! I know how many $5
and $10 bills you. You have 700 $5 bills, and
you have 200 $10 bills. The troll is
impressed, and he lets you cross the bridge to be the
hero or heroine of this fantasy adventure.