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# Systems of equations with substitution: shelves

Video transcript

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.