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Work, Rate, and Time in a System of Equations

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Today we're going to talk about solving problems involving work, rate, and time and systems of equations. So the relationship between work, rate, and time can be represented as w is equal to r times t where w is the amount of work that you do. For example, the amount of things that you complete, is going to be equal to how fast you can complete that work or the speed that you complete it multiplied by the time that you're doing the work for.

So we can rearrange this equation and solve it for different variables within the equation. For example, if I wanted to write this relationship in terms of time, I would isolate the t variable by dividing by r on both sides so I would see that time is equal to the work divided by the rate. Similarly, if I wanted to express this relationship in terms of the rate, to isolate my r variable, I would divide by my t variable on both sides. So that would give me that the rate is equal to the work divided by the time.

And another thing to know about solving problems involving work, rate, and time is that that the combined rate is going to be the sum of each individual rate. So for example, you could represent that as r1 plus r2 where r1 is the rate of work for one person and r2 is the rate of work for another person. So let's do an example again involving work, rate, and time for a real world scenario and see how we could represent it using a system of equations.

Suppose I have two people-- Kate and Jenny-- who work in a bicycle shop. Jenny can repair 48 bikes in the same amount of time that it takes Kate to repair 32 bikes. And we also know that Jenny can repair 2 more bikes per hour than Kate can. We want to know the rate that Jenny repairs bikes and the rate that Kate repairs bikes.

So let's start by defining our two variables for what we want to know. Let's let r1 be Jenny's rate for repairing bikes. And we'll let r2 be Kate's rate for repairing bikes. So the first thing that I know is that the amount of time that it takes Jenny to repair 48 bikes is the same as the amount of time that it takes Kate to repair 32 bikes.

So thinking about our relationship between work, rate, and time, I know that I can represent time as being equal to the work divided by the rate. So I'm going to define the amount of time that it takes Jenny to repair bikes as the work over her rate. So she can repair 48 bikes and that will be over her rate which is r1 and so this is equal to the amount of time for Jenny.

And I know that that's going to be equal to the amount of time for Kate. And she can repair 32 bikes and that will be over her rate, which is r2. So this can be the first equation for my system of equations.

And the second thing I know is that Jenny can repair 2 more bikes per hour than Kate can. So that means that Jenny's rate is going to be equal to whatever Kate's rate is plus 2. So this will be the second equation in my system of equations. So let's see how we can solve this to determine the rates for both Kate and Jenny.

All right. So we've got our two equations within our system. And because I have one of my variables in one of the equations are ready isolated, the substitution method is going to be the easiest method to use to solve this system. So I'm going to go ahead and substitute my expression that I have for r1 into my other equation for r1.

So now my first equation is going to become 48 over r2 plus 2 is equal to 32 over r2. I simply substituted what I had, my expression, for r1 into my first equation for r1. So now I can solve this equation because I only have one variable, r2, in the equation. Since this looks like a proportion, I'm going to solve it by cross multiplying.

So I'm going to have 48 times r2 is equal to 32 times r2 plus 2. I'm going to start by simplifying and distribute 32 times r2 plus 32 times 2 which is 64. Then I'm going to subtract 32r from both sides. Or 32r2 from both sides. And I'm left with 16 r2 is equal to 64. And finally, I'll divide both sides by 16. And I find that r2 is going to be equal to 4. So my rate for Kate is 4 bikes per hour.

So now I can use that value for r2 to find my value for r1 which is Jenny's rate for repairing bikes. And to do that, I'm going to use my second equation because it has what I'm looking for, the rate for Jenny already isolated. So that means that r1 is going to be equal to 4. That's what I just found for my value for r2 plus 2, which means r1 is equal to 6. So I found that Jenny can repair 6 bikes per hour, while Kate can repair 4 bikes per hour.

So finally, let's find the combined rate or the rate of both Jenny and Kate working together. So we know that the combined rate is just going to be the sum of the individual rates. And so that will be the sum of r1 and r2. I know that r1 or the rate that Jenny repairs bikes is 6 bikes per hour. And r2 or the rate that Kate repairs bikes is 4 bikes per hour. So 6 plus 4 is going to give me 10 bikes per hour, which is the combined rate that both Jenny and Kate can work.

So let's go over our key points from today. As usual, make sure you get them in your notes if you don't already so you can refer to them later. The amount of work completed is a product of the rate and time. Work can be a measure of the amount of things completed. This is related to the speed they're completed and the amount of time spent. And when rates work together, the combined rate is the sum of the individual rates.

So I hope that these key points and examples helped you understand a little bit more about solving problems involving work, rate, and time with systems of equations. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.