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CCSS.Math: , , ,

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 we are we 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 the system of equations and you will see that there are many ways of solving the systems of a system of equations but this time I want to do it visually because at least in my mind it helps really kind of 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 now let's say that this right over here this right over here is 500 tens that is 1,000 tens and let's say that this is I'm sorry that's 500 five that's one thousand five this is 500 tens 510s and this is one thousand tens so let's think about all of the combinations of F's and T's that satisfy this first equation if t if we have no tens then we're going to have nine hundred fives so if we have no tens we're going to have fought we have nine hundred five so that looks like it's right about there so that's the point zero tens 900 900 fives but what if it went the other way if we have no fives we're going to have 900 tens so it's going to be that's going to be the point nine hundred tens zero fives so all the combinations of F's and T's that satisfy this are going to be on this line are going to be on that line right over there and I dress draw a dotted line just because it's easier for me to draw it straight so that represents all the FS and T's that satisfy the first two 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 zero that we have five F is equal to fifty five hundred let me write do a little table here because this is a little bit more involved it's a little bit more involved so for the second equation tens and fives if I have no tens if I have no tens i5f is equal to 50 500 F will be 1,100 F is I must have 1,100 fives if I have no fives if I have no fives and this is 0 and F 10 T is equal to 5500 that means I have 550 550 10 so let's plot both of this point T equals 0 F is 1,100 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 C this would be 6 7 8 9 so 550 is going to be right over here so that is the point 550 550 comma 0 and all of these all of these points let me draw try to draw a straight line again I can do a better job than that I can do a better job than that so all of these points are the points I can let me try one more time one more time we want to get this right we don't want to be it pushed into the water by the troll so there you go that looks pretty good so alright every point on this blue line represents an F T combination that satisfies the second constraint so what is an F and T or number 5 s a 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 it is clearly on the yellow line and what we can do is if we drew this graph really 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 it 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 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 seven hundred fives 700 5's and it looks like we have about two hundred tens and this is based on my really rough graph but let's see if that worked seven hundred plus two hundred seven hundred plus two hundred is equal to nine hundred and if I have seven hundred five seven hundred let me write this down five times seven hundred is going to be the value of the fives which is thirty five hundred dollars and then ten plus ten times two hundred ten times 200 which is $2,000 $2,000 is the value of the tens 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 five and ten dollar bills you have you have seven hundred five dollar bills and you have two hundred ten dollar bills the troll is impressed and he lets you cross the bridge to be the hero or heroine of this fantasy adventure