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- We have the question: Devon is going to make 3 shelves for her father.
- She has a piece of lumber that is 12 feet long.
- She wants the top shelf to be half a foot shorter than the middle shelf.middle shelf.
- So let me do this in different colors.
- She wants the top shelf to be half a foot shorter than the middle shelf,
- let's just read the whole thing first,
- and the bottom shelf to be half a foot shorter than twice the length of the top shelf.
- Let me do that in a different color. I'll do that in blue.
- (Sal repeats) The bottom shelf to be half a foot shorter than twice the length of the top shelf.
- How long will each shelf be if she uses the entire 12 feet of wood?
- So let's define some variables for our different shelves,
- because that's what we have to figure out.
- We have the top shelf, the middle shelf, and the bottom shelf.
- So let's say that t is equal to length of top shelf-- t for top-- of top shelf.
- Let's make m equal the length of the middle shelf.
- m for middle.
- And then let's make b equal to the length of the bottom shelf
- b for bottom. Bottom shelf.
- So let's see what these different statements tell us.
- So this first statement, she says she wants the top shelf--
- and I'll do it in that same color--
- she wants the top shelf to be 1/2 a foot shorter than the middle shelf.
- So she wants the length of the top shelf to be--
- so this is equal to 1/2 a foot shorter than the middle shelf.
- So, if we're doing everything in feet,
- it's going to be the length of the middle shelf in feet minus 1/2, minus 1/2 feet.
- So that's what that sentence in orange is telling us.
- The top shelf needs to be 1/2 a foot shorter than the length of the middle shelf.
- Now, what does the next statement tell us?
- And the bottom shelf to be--
- so the bottom shelf needs to be equal to 1/2 a foot shorter than--
- so it's 1/2 a foot shorter than twice the length of the top shelf.
- So it's 1/2 a foot shorter than twice the length of the top shelf.
- These are the two statements interpreted in equal equation form.
- The top shelf's length has to be equal to the middle shelf's length minus 1/2.
- It's 1/2 foot shorter than the middle shelf.
- And the bottom shelf needs to be 1/2 a foot shorter than twice the length of the top shelf.
- And so how do we solve this?
- Well, you can't just solve it just with these two constraints,
- but they gave us more information.
- They tell us how long will each shelf be if she uses the entire 12 feet of wood?
- So the length of all of the shelves have to add up to 12 feet.
- She's using all of it.
- So t plus m, plus b needs to be equal to 12 feet.
- That's the length of each of them.
- She's using all 12 feet of the wood.
- So the lengths have to add to 12.
- So what can we do here?
- Well, we can get everything here in terms of one variable,
- maybe we'll do it in terms of m, and then substitute.
- So we already have t in terms of m.
- We could, everywhere we see a t, we could substitute
- with m minus 1/2.
- But here we have b in terms of t.
- So how can we put this in terms of m?
- Well, we know that t is equal to m minus 1/2.
- So let's take, everywhere we see a "t", let's substitute it with this thing right here.
- That is what t is equal to.
- So we can rewrite this blue equation as, the length of the
- bottom shelf is 2 times the length of the top shelf, t,
- but we know that t is equal to m minus 1/2.
- And if we wanted to simplify that a little bit,
- this would be that the bottom shelf is equal to--
- let's distribute the 2-- 2 times m is 2m.
- 2 times negative 1/2 is negative 1.
- And then minus another 1/2.
- Or, we could rewrite this as"
- b is equal to 2 times the middle shelf minus 3/2.
- Right?
- 1/2 is 2/2 minus another 1/2 is negative 3/2, just like that.
- So now we have everything in terms of m, and we can substitute back here.
- So the top shelf-- instead of putting a t there, we could put m minus 1/2.
- So we put m minus 1/2, plus the length of the middle shelf,
- plus the length of the bottom shelf.
- Well, we already put that in terms of m.
- That's what we just did.
- This is the length of the bottom shelf in terms of m.
- So instead of writing b there, we could write 2m minus 3/2.
- Plus 2m minus 3/2, and that is equal to 12.
- All we did is substitute for t.
- We wrote t in terms of m, and we wrote b in terms of m.
- Now let's combine the m terms and the constant terms.
- So if we have one m here, we have another m there,
- and then we have a 2m there.
- They're all positive.
- So 1 plus 1, plus 2 is 4m.
- So we have 4m.
- And then what do our constant terms tell us?
- We have a negative 1/2, and then we have a negative 3/2.
- So negative 1/2 minus 3/2, that is negative 4/2 or negative 2.
- So we have 4m minus 2.
- And, of course, we still have that equals 12.
- Now, we want to isolate just the m variable on one side of the equation.
- So let's add 2 to both sides to get rid of this 2 on the left-hand side.
- So if we add 2 to both sides of this equation, the
- left-hand side, we're just left with 4m-- these guys cancel out-- is equal to 14.
- Now, divide both sides by 4, we get m is equal to 14 over
- 4, or we could call that 7/2 feet, because we're doing everything in feet.
- So we solved for m,
- but now we still have to solve for t and b.
- So let's do that.
- Let's solve for t.
- t is equal to m minus 1/2.
- So it's equal to-- our m is 7/2 minus 1/2,
- which is equal to 6/2, or 3 feet.
- Everything is in feet, so that's how we know it's feet there.
- So that's the top shelf is 3 feet.
- The middle shelf is 7/2 feet, which is the same thing as 3 and 1/2 feet.
- And then the bottom shelf is 2 times the top shelf, minus 1/2.
- So what's that going to be equal to?
- That's going to be equal to 2 times 3 feet--
- that's what the length of the top shelf is-- minus 1/2, which is equal to 6 minus 1/2, or 5 and 1/2 feet.
- And we're done.
- And you can verify that these definitely do add up to 12.
- 5 and 1/2 plus 3 and 1/2 is 9, plus 3 is 12 feet, and it meets all of the other constraints.
- The top shelf is 1/2 a foot shorter than the middle shelf,
- and the bottom shelf is 1/2 a foot shorter than 2 times the top shelf.
- And we are done.
- We know the lengths of the shelves that Devon needs to make.

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