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A jogger and a walker set out at 9:00 AM from the same

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point, headed in the same direction.

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The average speed of the jogger is 1 mile per hour

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slower than twice the speed of the walker.

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So that's going to be an equation right there.

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The average speed of the jogger is 1 mile per hour

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slower than twice the speed of the walker.

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And then they say, in 2 hours, the jogger is 3 miles ahead of

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the walker.

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That's going to be another equation.

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In 2 hours, the jogger is 3 miles ahead of the walker.

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Find the rate of the jogger.

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So let's write down that first statement in equation form.

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The average speed of the jogger, let's call that the

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speed of the jogger.

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That's the average speed of the jogger is 1 mile per hour

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slower than twice the speed of the walker.

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So that's equal to 2 times the speed of the

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walker, minus 1, right?

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It's 1 mile per hour slower than twice the

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speed of the walker.

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So that's that first statement.

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And then they say, in 2 hours, the jogger is 3 miles ahead of

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the walker.

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So if we go 2 hours into the future.

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So in 2 hours, the second statement is the jogger is 3

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miles ahead of the walker.

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So we could say that the distance that the jogger has

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traveled, so the distance of the jogger, is 3 miles more

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than the distance of the walker.

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It is equal to the distance of the walker plus 3.

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That's all I'm saying.

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This is how far the jogger's traveled.

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He's traveled 3 miles more than the walker.

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Now right now I have two equations with four unknowns.

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That's not cool.

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But they've given us some information.

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They said this has happened after 2 hours.

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So we can express the distances they've traveled in

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terms of the speed they traveled at.

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Just as a bit of a review, distance is equal to rate

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times time.

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In this situation, the time is 2.

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The rate is the speed of each person.

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So we can figure out the distance.

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So if you want to know the distance that the jogger

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traveled, that's just going to be 2 hours times the speed of

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the jogger.

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So it's just going to be 2 times the speed

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of the jogger, right?

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Distance is equal to rate times time,

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or time times rate.

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Either way.

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What's the distance of the walker?

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Well, it's going to be 2 hours times the speed

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of the walker, right?

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Distance is equal to rate times time.

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So now we can rewrite this equation in terms of the

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speeds of the jogger and the walker.

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So this equation right here.

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We can rewrite it as, instead of writing the distance of the

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jogger, we could write 2 times the speed of the jogger--

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that's that right there-- is equal to-- instead of writing

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the distance of the walker, we could write 2 times the speed

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of the walker, make sure that's a subscript-- speed of

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the walker plus 3.

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Now we have two equations and two unknowns, and it looks

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pretty tempting to eliminate one of the variables.

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In particular, we have a 2 times the speed of the walker,

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we have a 2 times the speed of the walker here.

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What we can do is we can multiply this second equation

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by negative 1 and then add the 2's.

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Then these guys will cancel out.

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So let's do that.

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Actually let's do it even a better way.

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Let's multiply the top equation by negative 1 and

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then add them.

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So let's multiply the top equation by negative 1.

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This becomes a negative times the speed of the jogger.

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This becomes a negative 2 times the speed of the walker.

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Negative 1 times negative 1 times negative 1 is plus 1.

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And now if we add these two equations.

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These two guys are going to cancel out.

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And what do we get?

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We get 2 times the speed of the jogger minus the speed of

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the jogger.

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Well that's just going to be the speed

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of the jogger, right?

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2 of something minus 1 of something is

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just 1 of that thing.

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These two guys cancel out.

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That was the whole point behind multiplying the top

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equation by negative 1.

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Then you have 1 plus 3, so the speed of the jogger is 4.

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And we're done.

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That's all they want us to find.

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Find the rate of the jogger.

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And we just figured it out.

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If we also wanted to figure out the rate of the walker, we

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could substitute back into this top equation right here,

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or the original version of it, which was the speed of the

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jogger is equal to 2 times the speed of the walker minus 1.

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We know the speed of the jogger is 4, so you get 4 is

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equal to 2 times the speed of the walker minus 1.

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Add 1 to both sides, you get 5 is equal to 2 times the speed

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of the walker.

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Divide both sides by 2, and you get the speed of the

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walker is 2-1/2 miles per hour.

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But we didn't even have to do that.

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This was the answer that the problem was looking for.

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