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Jason bicycled from home to the train station at an

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average speed of 10 miles per hour.

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Then he boarded a train and traveled into the city at an

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average speed of 50 miles per hour.

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The entire distance was 30 miles; the

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entire trip took 1 hour.

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How many miles did Jason travel by train?

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So let's write that down as a variable.

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So let's say distance, and we'll write a little small t

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right here, a little subscript t.

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This is the distance by train, so train distance.

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Right there.

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And let's have a d with a little bit of a

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little b right there.

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That is the bike distance.

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Now, they give us one piece of information, that the total

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distance, the entire distance, was 30 miles.

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So that means that the distance by train plus the

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distance by bike is 30 miles.

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So I could write that right here.

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The distance by train plus the distance by bike

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is equal to 30 miles.

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That's what that constraint tells us.

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Now the next one, they tell us that the

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entire trip took 1 hour.

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So the time by train plus the time by bike took 1 hour.

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And you might be thinking, hey, gee, that's going to

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introduce two new variables.

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But let's think a little bit about whether we can express

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the time by train and the time by bike in terms

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of these two variables.

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So just as a bit of review-- I think this is review-- you're

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probably familiar with distance is equal to rate

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times time.

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Or if you divide both sides of this equation by rate, you get

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time is equal to distance divided by rate.

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So let's think about the situation for

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each of these guys.

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What is the time by train?

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I'll write it like this.

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The time by train is going to be equal to the distance by

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train divided by the rate that the train was going at.

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And they gave us that information.

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They told us that the train traveled into the city at an

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average speed of 50 miles per hour.

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That is the rate of the train.

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So the time of the train was the distance of the train

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divided by 50.

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Same exact argument.

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The time of the bicycle, the time traveled on the bicycle,

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will be the distance traveled on the bicycle divided by the

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speed of the bicycle.

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And they give us the speed right there,

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10 miles per hour.

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And everything in this problem we're assuming is going to be

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in either miles or hours.

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There's not any major unit conversion.

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So that's there right there.

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So the second statement is that the

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entire trip took 1 hour.

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So the time by train plus the time by bicycle is

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going to be 1 hour.

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Actually, let me do that in a different color.

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The entire trip took 1 hour.

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So this time plus this time is 1 hour.

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And I'm going to write it in terms of the distances so I

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only have two unknowns.

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So the time by train is the distance by train divided by

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50 plus the time by bicycle is the distance by bicycle

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divided by 10.

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And then that is equal to 1 hour.

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That's what this statement right here is telling us,

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although we had to use that information and that

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information to divide by the 10 and to divide by the 50.

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Now we have two equations of two unknowns.

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And our whole goal, the whole purpose of this problem, they

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want to know how many miles did Jason travel by train.

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So we want to figure out that variable, or we want to figure

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out that variable.

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Now, the easiest way to do this is if we can just

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eliminate the distance by bicycle and then solve for the

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distance by train.

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Then we're done.

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We would have solved the problem.

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Now, the easiest way I can think of doing that is this

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one is pretty simple already.

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Let me just rewrite it to the right here.

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So the distance by train plus the distance by bicycle is

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equal to 30.

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And I want to cancel out the distance by bicycle.

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So if I could make this into just a negative distance by

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bicycle, then I'm all set.

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And if I add the two equations, they'll cancel out.

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So the easiest way to just make this a negative distance

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by bicycle is multiply both sides of this equation by

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negative 10.

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Because you multiply negative 10 times this, the 10's cancel

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out, this just becomes a negative distance by bicycle.

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So let's write it over here.

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Negative 10 times the distance by train over 50.

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10 divided by 50 is 1/5, so it's negative

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1/5 distance by train.

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And then negative 10 times this expression right here,

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the 10's cancel out.

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It's just negative distance by bicycle is equal to-- forgot

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that parentheses-- is equal to negative 10.

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The whole point here was so that this becomes the negative

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version of that.

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So when I add these two equations, they'll cancel out.

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So let's do that.

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Let's add these two equations.

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So if you add the left-hand side, these guys cancel out.

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You have 1 dt minus 1/5 dt.

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1 dt you can view as 5/5 dt.

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So 5/5 minus 1/5, you have 4/5 distance by train is equal to

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30 plus negative 10, or is equal to 20.

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Now to solve for distance by train, we can just multiply

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both sides of this expression, both sides of this equation,

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by the inverse of 4/5.

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So we'll multiply both sides by 5/4.

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The whole point of that is that this cancels out, and we

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are left with the distance by train is equal to 20 times

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5/4, that's the same thing as 20/1 times 5/4.

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20 divided by 4 is 5, 4 divided by 4 is 1.

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So it's just 5 times 5.

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So the distance traveled by train is 25 miles.

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And we're done.

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And we could go back if we wanted to figure out any of

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the times, or the distance by bicycle.

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Actually, it's very easy.

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The distance by train is 25, distance by bicycle has to be

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5 if they're going to add up to be 30 miles.

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