Hi, and welcome. My name is Anthony Varela. And today we're going to be solving a system of equations that represents work, rate, and time. So first we're going to talk about this relationship between work, rate, and time. This will help us then develop equations that make up a system, and then we'll go ahead and solve that system of equations. So first, let's talk about work, rate, and time, and let's put this within a context of packaging boxes.
Now suppose I know that in one hour I can package 15 boxes. So this then describes the rate at which I pack boxes. Now if I work, then, for two hours packaging boxes, this would represent a time. And I know that in those two hours, then, I packaged 30 boxes, and I'm going to call this, then, the work that I've done.
Now how did I get to 30 boxes? Well I multiplied 15 boxes per hour, the rate, by two hours, the time, to get 30 boxes. So here, we could say, then, that work equals rate times time. So I'm going to abbreviate this, w equals r times t. And we're going to be using this equation to develop a system of equations and then solve for some variables.
So going with this packaging boxes scenario, I'm going to give you some pieces of information and then we'll answer a question here. So we know that Alice can package 70 boxes in some period of time. We're not quite sure what that time is. But in the same amount of time, Brady packages 60 boxes. And we also know that Alice packages three more per hour than Brady does. So what we want to know is how many boxes can Alice and Brady package in one hour if they're working together?
So first, let's talk about working together. So when we have people working together, we combine their rates. So we can say that r sub 1 is the rate of one person, r sub 2 is the rate of a second person. And if they're working together, we add those rates together. So we'll come back to that in a minute.
But what we want to do is to set up a system of equations, which means first we have to define some variables. Well, if I can solve for Alice's rate and solve for Brady's rate, I can add those two together, and then just multiply by one hour to find out how many boxes they could package in one hour. So I'm going to define my variables as Alice's rate and Brady's rate. So let's say x equals Alice's rate. This is going to be in boxes per hour. And y equals Brady's rate, also measured in boxes per hour.
So now that we have some defined variables, let's develop a system of equations. And we're going to be using work equals rate times time. So what I know about Alice is I know that she packs 70 boxes, and that is a product of her rate, x, multiplied by some period of time. So 70 equals x times t. Now we're going to develop a very similar equation to represent Brady's work, rate, and time. So he can do 60 boxes, and that is a product of his rate, y, also multiplied by t, time.
So here are two of our equations that make up our system. We actually know another piece of information worth writing down. We know that Alice can do three more boxes per hour than Brady. So I'm going to say that x, Alice's rate, is 3 plus y. So that would be Brady's rate plus 3. So here is the equations that I know-- 70 equals x times t, 60 equals y times t, and x equals 3 plus y. How are we going to solve for x, and how are we going to solve for y?
Well, we're going to be solving this by substitution. And substitution is really just a whole bunch of rewriting and replacing. So we're going to write equivalent equations, and then we're going to substitute by putting in equivalent expressions in for some variables. Now they already have a really good head start.
Usually what I do when I solve by substitution is I try to rewrite as x equals or y equals. And I actually already have x equals 3 plus y. So what I'm going to do is take my equation that has x in it, so 70 equals x times t, and I'm going to write in 3 plus y instead of x. So I've substituted that equivalent expression for x.
Well how does this help me? I'm going to distribute the t into 3 and y. So 70 equals 3t plus yt. You still might be thinking, how does this help? Usually at this point I have a single variable equation. I still have two variables, time and Brady's rate.
Well, I know that yt equals 60. I got that from Brady packaging 60 boxes in some number of hours. So I'm actually going to replace, then, yt with 60. So I have 70 equals 3t plus 60. So if I take away, then, 60 from both sides of this equation, I have 10 equals 3t, so I can solve for time. Time is 10/3 hours, right? So that's good news. I've solved for one of my variables. So I can now replace t with 10/3.
So I have 70 equals x times t, 60 equals y times t, x equals 3 plus y, and I know that t equals 10/3. So here's where I have t in my first two equations, so let's go ahead and replace t with 10/3 in both our first equation and our second equation. This makes it super easy, then, to solve for both x and y.
So I'm going to multiply by that denominator 3, so I have 210 equals 10x, and 180 equals 10y. And now I can divide by 10, so I know that x equals 21, and y equals 18. So I know that Alice can package boxes at 21 boxes per hour, and I know that Brady can do 18 in one hour.
So what was my original question? How many can they package in one hour if they're working together? So working together, I'm going to add those two rates. So 21 plus 18 is 39. So in one hour, Alison and Brady can package 39 boxes together.
So let's review work, rate, and time and a system of equations. Well, we have a relationship between work, rate, and time. Work equals rate times time. We talked about two people working together, so this would be a combined rate-- the rate of one person plus the rate of a second person. We created a system of equations by first defining variables from our situation and using those variable definitions to develop equations that make up a system, and we solve this by substitution.
So we rewrote equivalent equations, substituted those expressions in for variables. This allowed us to solve a single variable equation. And once we solve for one variable, we can keep on substituting that back into our original equations to solve for our other variables. So thanks for watching this tutorial on work, rate, and time using a system of equations. Hope to see you next time.