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So we have the question, Devon is going to make 3 shelves for

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her father.

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She has a piece of lumber that is 12 feet long.

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She wants the top shelf to be half a foot shorter than the

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middle shelf.

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So let me do this in different colors.

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She wants the top shelf to be half a foot shorter than the

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middle shelf.

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So I'll just-- let's just read the whole thing first. And the

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bottom shelf to be half a foot shorter than twice the length

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of the top shelf.

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Let me do that in a different color.

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I'll do that in blue.

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The bottom shelf to be half a foot shorter than twice the

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length of the top shelf.

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How long will each shelf be if she uses the

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entire 12 feet of wood?

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So let's define some variables for our different shelves,

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because that's what we have to figure out.

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We have the top shelf, the middle shelf,

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and the bottom shelf.

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So let's say that t is equal to length of top shelf-- t for

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top-- of top shelf.

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Let's make m equal the length of the middle shelf.

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m for middle.

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And then let's make b equal to the length of the bottom

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shelf-- b for bottom-- bottom shelf.

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So let's see what these different statements tell us.

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So this first statement, she says she wants the top shelf--

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and I'll do it in that same color-- she wants the top

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shelf to be 1/2 a foot shorter than the middle shelf.

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So she wants the length of the top shelf to be-- so this is

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equal to 1/2 a foot shorter than the middle shelf.

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So, if we're doing everything in feet, it's going to be the

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length of the middle shelf in feet minus

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1/2, minus 1/2 feet.

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So that's what that sentence in orange is telling us.

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The top shelf needs to be 1/2 a foot shorter than the length

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of the middle shelf.

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Now, what does the next statement tell us?

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And the bottom shelf to be-- so the bottom shelf needs to

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be equal to 1/2 a foot shorter than-- so it's 1/2 a foot

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shorter than twice the length of the top shelf.

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So it's 1/2 a foot shorter than twice the

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length of the top shelf.

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These are the two statements interpreted in

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equal equation form.

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The top shelf's length has to be equal to the middle shelf's

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length minus 1/2.

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It's 1/2 foot shorter than the middle shelf.

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And the bottom shelf needs to be 1/2 a foot shorter than

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twice the length of the top shelf.

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And so how do we solve this?

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Well, you can't just solve it just with these two

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constraints, but they gave us more information.

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They tell us how long will each shelf be if she uses the

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entire 12 feet of wood?

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So the length of all of the shelves have to

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add up to 12 feet.

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She's using all of it.

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So t plus m, plus b needs to be equal to 12 feet.

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That's the length of each of them.

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She's using all 12 feet of the wood.

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So the lengths have to add to 12.

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So what can we do here?

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Well, we can get everything here in terms of one variable,

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maybe we'll do it in terms of m, and then substitute.

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So we already have t in terms of m.

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We could, everywhere we see a t, we could substitute

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with m minus 1/2.

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But here we have b in terms of t.

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So how can we put this in terms of m?

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Well, we know that t is equal to m minus 1/2.

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So let's take, everywhere we see a t, let's substitute it

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with this thing right here.

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That is what t is equal to.

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So we can rewrite this blue equation as, the length of the

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bottom shelf is 2 times the length of the top shelf, t,

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but we know that t is equal to m minus 1/2.

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And if we wanted to simplify that a little bit, this would

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be that the bottom shelf is equal to-- let's distribute

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the 2-- 2 times m is 2m.

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2 times negative 1/2 is negative 1.

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And then minus another 1/2.

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Or, we could rewrite this as b is equal to 2 times the middle

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shelf minus 3/2.

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Right?

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1/2 is 2/2 minus another 1/2 is negative

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3/2, just like that.

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So now we have everything in terms of m, and we can

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substitute back here.

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So the top shelf-- instead of putting a t there, we could

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put m minus 1/2.

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So we put m minus 1/2, plus the length of the middle

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shelf, plus the length of the bottom shelf.

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Well, we already put that in terms of m.

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That's what we just did.

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This is the length of the bottom shelf in terms of m.

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So instead of writing b there, we could write 2m minus 3/2.

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Plus 2m minus 3/2, and that is equal to 12.

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All we did is substitute for t.

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We wrote t in terms of m, and we wrote b in terms of m.

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Now let's combine the m terms and the constant terms. So if

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we have, we have one m here, we have another m there, and

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then we have a 2m there.

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They're all positive.

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So 1 plus 1, plus 2 is 4m.

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So we have 4m.

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And then what do our constant terms tell us?

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We have a negative 1/2, and then we have a negative 3/2.

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So negative 1/2 minus 3/2, that is negative 4/2 or

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negative 2.

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So we have 4m minus 2.

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And, of course, we still have that equals 12.

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Now, we want to isolate just the m variable on one side of

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the equation.

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So let's add 2 to both sides to get rid of this 2 on the

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left-hand side.

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So if we add 2 to both sides of this equation, the

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left-hand side, we're just left with 4m-- these guys

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cancel out-- is equal to 14.

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Now, divide both sides by 4, we get m is equal to 14 over

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4, or we could call that 7/2 feet, because we're doing

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everything in feet.

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So we solved for m, but now we still have to

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solve for t and b.

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So let's do that.

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Let's solve for t.

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t is equal to m minus 1/2.

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So it's equal to-- our m is 7/2 minus 1/2, which is equal

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to 6/2, or 3 feet.

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Everything is in feet, so that's how we

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know it's feet there.

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So that's the top shelf is 3 feet.

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The middle shelf is 7/2 feet, which is the same thing as 3

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and 1/2 feet.

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And then the bottom shelf is 2 times the top

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shelf, minus 1/2.

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So what's that going to be equal to?

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That's going to be equal to 2 times 3 feet-- that's what the

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length of the top shelf is-- minus 1/2, which is equal to 6

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minus 1/2, or 5 and 1/2 feet.

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And we're done.

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And you can verify that these definitely do add up to 12.

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5 and 1/2 plus 3 and 1/2 is 9, plus 3 is 12 feet, and it

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meets all of the other constraints.

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The top shelf is 1/2 a foot shorter than the middle shelf,

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and the bottom shelf is 1/2 a foot shorter than 2

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times the top shelf.

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And we are done.

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We know the lengths of the shelves that

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Devon needs to make.