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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) ?
(43 votes)
• Also by matrices after some more levels of math practice.
(6 votes)
• imagine being the troll waiting as sal explains your riddle
(26 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?
(6 votes)
• Day in the life of a kid who is forced to study during the summer...
(18 votes)
• real.
(3 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.
(15 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
(16 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?
(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.
(18 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 would be 901 bills, and he only has 900. Close but no cigar!
(4 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.
(2 votes)
• 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.