Introduction to systems of equations
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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.