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## 8th grade (Eureka Math/EngageNY)

### Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions- 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
- Systems of equations with graphing
- Systems of equations with graphing: 5x+3y=7 & 3x-2y=8
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations with graphing: chores
- Systems of equations with graphing
- Systems of equations with elimination: 3t+4g=6 & -6t+g=6
- Systems of equations with elimination
- Systems of equations with elimination: x+2y=6 & 4x-2y=14
- Systems of equations with elimination: -3y+4x=11 & y+2x=13
- Systems of equations with elimination: 2x-y=14 & -6x+3y=-42
- Systems of equations with elimination: 4x-2y=5 & 2x-y=2.5
- Systems of equations with elimination: x-4y=-18 & -x+3y=11
- Systems of equations with elimination
- Systems of equations with elimination: 6x-6y=-24 & -5x-5y=-60
- Systems of equations with elimination challenge
- Systems of equations with substitution: 2y=x+7 & x=y-4
- Systems of equations with substitution
- Systems of equations with substitution: y=4x-17.5 & y+2x=6.5
- Systems of equations with substitution: -3x-4y=-2 & y=2x-5
- Systems of equations with substitution: 9x+3y=15 & y-x=5
- Systems of equations with substitution
- Systems of equations with substitution: y=-5x+8 & 10x+2y=-2
- Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120
- Systems of equations with elimination: apples and oranges
- Systems of equations with elimination: TV & DVD
- Systems of equations with elimination: King's cupcakes
- Systems of equations with elimination: Sum/difference of numbers
- Systems of equations with elimination: potato chips
- Systems of equations with elimination: coffee and croissants
- Systems of equations with substitution: coins
- Systems of equations with substitution: potato chips
- Systems of equations with substitution: shelves
- Systems of equations word problems
- Age word problem: Imran
- Age word problem: Ben & William
- Age word problem: Arman & Diya
- Age word problems
- Solutions to systems of equations: consistent vs. inconsistent
- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Systems of equations number of solutions: y=3x+1 & 2y+4=6x
- Solutions to systems of equations: dependent vs. independent
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Forming systems of equations with different numbers of solutions
- Number of solutions to a system of equations algebraically
- Comparing Celsius and Fahrenheit temperature scales
- Converting Fahrenheit to Celsius

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# 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) ?(40 votes)
- Also by matrices after some more levels of math practice.(4 votes)

- Is there another method to find the answer than drawing the graphs?(1 vote)
- 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.(4 votes)

- can this be solved without graph?(1 vote)
- 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.(17 votes)

- imagine being the troll waiting as sal explains your riddle(8 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.(7 votes)
- 702+199 would be 901 bills, and he only has 900. Close but no cigar!(2 votes)

- How are they typing with a mouse! I can't even use a pencil correctly(7 votes)
- I assume Sal uses a pen tablet, which makes it much easier to draw.(2 votes)

- I just inserted my own values and got the right answer, is that still a good option. It did not take me that long probably like 2 minutes. Is this strategy too inconsistent. I got 700 5´s and 200 10´s. I inserted values that add up to 900 and checked if it equaled 5500 I started big and went down to increments of 50 each time if the values I inserted were wrong. I just need to know if this work for me in the future, is it?(2 votes)
- This method is called either trial and error or how you present it more like an educated guess, so it will work, but I do not know if it will always be as fast. The biggest issues will probably be when the answer is a decimal or fraction.(5 votes)

- What would you do if you have multiple variables?

Like if the troll takes 5 dollar bills 10 dollar bills AND 20 dollar bills.

How would you solve that?

If somebody could explain it step-by-step that would be really helpful.

Thank You!(2 votes)- I could solve it but first i need to know how much dollor bills he took in total.(3 votes)

- I solved it before I finished the first video

there are 700 $5 bills and 200 $10 bills

first i divided 5500 by 10 and 5

10 = 550

5 = 1100

1100 - 900 = 200

900 - 200 = 700

700(5) + 200(10) = 5500

is this a way to do it?

also it took Sal over 10 minutes, he would have been pushed into the water, or eaten like trolls normally do to the trespassers of there bridges.(4 votes) - i did the math and you solved the riddle in approximately 9 minutes and 50 seconds (I rounded down). That was close, you nearly died.(3 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.