Systems of equations with substitution: shelves

So 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. So let me do this in different colors. She wants the top shelf to be half a foot shorter than the middle shelf. So I'll just-- 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. 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, 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.