Systems of equations word problems
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Let's say that Arman today is 18 years old. And let's say that Diya today is 2 years old. And what I am curious about in this video is how many years will it take-- and let me write this down-- how many years will it take for Arman to be three times as old as Diya? So that's the question right there, and I encourage you to try to take a shot at this yourself. So let's think about this a little bit. We're asking how many years will it take. That's what we don't know. That's what we're curious about. How many years will it take for Arman to be three times as old as Diya? So let's set some variable-- let's say, y for years. Let's say y is equal to years it will take. So given that, can we now set up an equation, given this information, to figure out how many years it will take for Arman to be three times as old as Diya? Well, let's think about how old Arman will be in y years. How old will he be? Let me write here. In y years, Arman is going to be how old? Arman is going to be-- well, he's 18 right now-- and in y years, he's going to be y years older. So in y years, Arman is going to be 18 plus y. And what about Diya? How old will she be in y years? Well, she's 2 now, and in y years, she will just be 2 plus y. So what we're curious about, now that we know this, is how many years will it take for this quantity, for this expression, to be three times this quantity? So we're really curious. We want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y. Notice, this is Arman in y years. This is Diya in y years. And we're saying that what Arman's going to be in y years is three times what Diya is going to be in y years. So we've set up our equation. Now we can just solve it. So let's take this step by step. So the left hand side-- and maybe I'll do this in a new color, just so I don't have to keep switching-- so on the left hand side, I still have 18 plus y. And on the right hand side, I can distribute this 3. So 3 times 2 is 6. 3 times y is 3y. 6 plus 3y. And then it's always nice to get all of our constants on one side of the equation, all of our variables on the other side of the equation. So we have a 3y over here. We have more y's on the right hand side than the left hand side. So let's get rid of the y's on the left hand side. You could do it either way, but you'd end up with negative numbers. So let's subtract a y from each side. And we are left with, on the left hand side, 18. And on the right hand side you have 6 plus 3 y's. Take away one of those y's. You're going to be left with 2 y's. Now we can get rid of the constant term here. So we will subtract 6 from both sides. 18 minus 6 is 12. The whole reason why we subtracted 6 from the right was to get rid of this, 6 minus 6 is 0, so you have 12 is equal to 2y. Two times the number of years it will take is 12, and you could probably solve this in your head. But if we just want a one-coefficient year, we would divide by 2 on the right. Whatever we do to one side of an equation, we have to do it on the other side. Otherwise, the equation will not still be an equation. So we're left with y is equal to 6, or y is equal to 6. So going back to the question, how many years will it take for Arman to be three times as old as Diya? Well, it's going to take six years. Now, I want you to verify this. Think about it. Is this actually true? Well, in six years, how old is Arman going to be? He's going to be 18 plus 6. We now know that this thing is 6. So in six years, Arman is going to be 18 plus 6, which is 24 years old. How old is Diya going to be? Well, she's going to be 2 plus 6, which is 8 years old. And lo and behold, 24 is, indeed, three times as old as 8. In 6 years-- Arman is 24, Diya is 8-- Arman is three times as old as Diya, and we are done.