Main content

### Course: Algebra 1 > Unit 6

Lesson 1: Introduction to systems of equations- Systems of equations: trolls, tolls (1 of 2)
- Systems of equations: trolls, tolls (2 of 2)
- Testing a solution to a system of equations
- Solutions of systems of equations
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations with graphing: exact & approximate solutions
- Systems of equations with graphing
- Setting up a system of equations from context example (pet weights)
- Setting up a system of linear equations example (weight and price)
- Creating systems in context
- Interpreting points in context of graphs of systems
- Interpret points relative to a system

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

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

Solving the system of equations visually. Now we can save the prince/princess. Created by Sal Khan.

## Want to join the conversation?

- I was wondering how many different ways can you solve this question (the one it asks in the video) ?(51 votes)
- Also by matrices after some more levels of math practice.(7 votes)

- In the last video sal said that you have to solve the riddle in under 10 minutes, it takes 11 for Sal to solve it. So we are drowning in the river(43 votes)
- Watch the video in faster playback then you could warp time and the troll won't drown you.(8 votes)

- imagine being the troll waiting as sal explains your riddle(39 votes)
- Troll:

Hmm...

Oh... So that's how you solve it...

Should I just throw him off the bridge...

It takes too much time...

Oh is it about to end?(15 votes)

- Is there another method to find the answer than drawing the graphs?(3 votes)
- Well, yes.

Assume that all of the 900 dollar bills are all 5 dollars bills.

-> the total amount of money is 900*5=4500 dollar

However, the real amount of money is $5500. Therefore, the difference is 5500-4500=1000 dollar.

The difference between a $5 bill and a $10 bill is $5.

-> the number of $10 bill is 1000/5=200 bills.

-> the number of $5 bill is 900-200=700 bills.(25 votes)

- Could someone elaborate on this a little bit more? It seems to me like a guess work. Isn't it just as feasible that the troll has 702 $5 bills and 199 $10 bills? If not, why not? How does the two graphs intersecting prove the different configurations of $10 and $5 bills that the troll has? I need someone to walk me through the logical reasoning and steps behind this.(8 votes)
- 702 + 199 = 901 bills instead of 900 as required by the problem. If you try 701 and 199 which does equal 900 bills, the money woud be 701*5 + 199*10 = 5495, and 699 and 201 would give 699*5 + 201*10 = 5505, both of which are close to 5500, but not exact.

The idea is that for every variable you have, in order to find a unique solution, you have to have a unique equation (not the same slope) for the number of variables. The place of intersection is a value of (x,y) that uniquely works for both equation, the only point that is on both lines.(19 votes)

- sal has godlike drawing skills(18 votes)
- I am here by myself. Look, math is a vital part in life and it can be required in numerous fields. Thus, if you want to be a great specialist you need math.(12 votes)
- can this be solved without graph?(0 votes)
- Yes, systems can be solved graphically, but they can also be solved using methods of elimination, substitution, or a combination of both. Very rarely will graphically solving be handy because of human errors when drawing and the time it takes to graph.(22 votes)

- How are they typing with a mouse! I can't even use a pencil correctly(8 votes)
- I assume Sal uses a pen tablet, which makes it much easier to draw.(1 vote)

- Is there a method to do this which doesn't require plotting or guesstimating?(3 votes)
- Yes, there are 2 other methods: 1) substitution method; 2) elimination method.

Graphing method is often taught first to help you understand the types of solutions and what they mean in terms of how the 2 lines relation to each other.

-- Intersection lines: 1 solution which is the point of intersection.

-- Parallel lines: No solution because the lines never touch each other.

-- Same line: Solution set is all the points on the line.

Then, the other two methods are taught that provide precise solutions.(5 votes)

## 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.