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## Equivalent systems of equations and the elimination method

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# Systems of equations with elimination: King'sÂ cupcakes

CCSS Math: HSA.REI.C.6

## Video transcript

After you cross the troll's
bridge and you save the prince or princess, you return them
back to their father, the King. And he's so excited that you
returned their son or daughter to him that he wants to
throw a brunch in your honor. But he has a little
bit of a conundrum in throwing the brunch. He wants to figure out how
many cupcakes should he order? He doesn't want to
waste any, but he wants to make sure that
everyone has enough to eat. And you say, well
what's the problem here? And he says, well I know
adults eat a different number of cupcakes than children eat. And I know that in my
kingdom, an adult will always eat the same amount
and a child will always eat the same amount. And so you say, King, well what
information can you give me? I might be able to help
you out a little bit. You're feeling very confident
after this troll situation. And he says, well I know at the
last party we had 500 adults and we had 200
children, and combined they ate 2,900 cupcakes. And you say, OK,
that's interesting. But I think I'll need a
little bit more information. Have you thrown
parties before then? And the king says,
of course I have. I like to throw parties. Well what happened at
the party before that? And he says, well there
we also had 500 adults and we had 300 children. And you say, well
how many cupcakes were eaten at that party? And he says, well we know
it was 3,100 cupcakes. And so you get a
tingling feeling that a little bit of algebra
might apply over here. And you say, well let me see. What do we need to figure out? We need to figure out the
number of cupcakes on average that an adult will eat. So number of cupcakes
for an adult. And we also need to
figure out the number of cupcakes for a child. So these are the
two things that we need to figure out
because then we can know how many adults
and children are coming to the next brunch that are
being held in your honor and get the exact right
number of cupcakes. So those are things you're
trying to figure out. And we don't know
what those things are. Let's define some variables
that represent those things. Well let's do a for adults. Let's let a equal the
number of cupcakes that, on average,
each adult eats. And let's do c for children. So c is the number of cupcakes
for a child on average. So given that
information, let's see how we can represent what the
King has told us algebraically. So let's think about this
orange information first. How could we represent
this algebraically? Well let's think about
how many cupcakes the adults ate at that party. You had 500 adults, and
on average, each of them exactly a cupcakes. So the total number of
cupcakes that the adults ate were 500 times a. How many did the children eat? Well same logic. You had 200 children and
they each ate c cupcakes. So 200 times c is the
total number of cupcakes that the children ate. Well how much did
they eat in total? Well it's the total
number that the adults ate plus the total number
that the children ate which is 2,900 cupcakes. So let's do that and apply that
same logic to the blue party right over here, this
blue information. How can we represent
this algebraically? Well once again, how many total
cupcakes did the adults eat? Well, you had 500
adults and they each ate a cupcakes, which
is an unknown right now. And then what
about the children? Well you had 300 children
and they each ate c cupcakes. And so if you add
up all the cupcakes that the adults ate plus all the
cupcakes that the children ate, you get to 3,100 cupcakes. So this is starting
to look interesting. I have two equations. I have a system of two
equations with two unknowns. And you know from your
experience with the troll that you should be
able to solve this. You could solve it graphically
like you did in the past, but now you feel
that there could be another tool in your
tool kit which is really just an application of the
algebra that you already know. So think a little bit about
how you might do this. So let's rewrite this first
equation right over here. So we have 500a plus 200c
is equal to 2,900 cupcakes. Now, it would be good if we
could get rid of this 500a somehow. Well you might say, well
let me just subtract 500a. So you might say, oh, I
just want to subtract 500a. But if you subtracted 500a
from the left hand side, you'd also have to subtract
500a from the right hand side. And so the a wouldn't
just disappear. It would just end up
on the right hand side, and you would still
have one equation with two unknowns which
isn't too helpful. But you see something
interesting. You're like, well
this is a 500a here. What if I subtracted
a 500a and this 300c? So if I subtracted
the 500a and the 300c from the left hand side. And you're like, well
why is that useful? You're going to do the same
thing on the right hand side and then you're going to have
an a and a c on the right hand. And you just say, hold on,
hold on one second here. Hold on, I guess you're
talking to yourself. Hold on one second. I'm subtracting the left
hand side of this equation, but this left hand side
is the exact same thing as this right hand side. So here I could
subtract 500a and 300c, and I could do 500a and
subtract 300c over here. But we know that
subtracting 500a and 300c, that's the exact same
thing as subtracting 3,100. Let me make it clear. This is 500a minus 300c
is the exact same thing as subtracting 500a plus 300c. And we know that 500a plus
300c is exactly 3,100. This is 3,100. This is what the second
information gave us. So instead of subtracting
500a and minus 300c, we can just subtract 3,100. So let me do that. This is exciting. So let me clear that out. So let's clear that out. And so here, instead
of doing this, I can subtract the exact same
value, which we know is 3,100, subtract 3,100. So looking at it
this way, it looks like we're subtracting
this bottom equation from the top equation, but
we're really just subtracting the same thing from both sides. This is just very
basic algebra here. But if we do that,
let's see what happens. So on the left hand side, 500a
minus 500a, those cancel out. 200c minus 300c, that
gives us negative 100c. And on the right hand
side, 2,900 minus 3,100 is negative 200. Well now we have one
equation with one unknown, and we know how to solve this. We can divide both
sides by negative 100. These cancel out. And then over here, you
end up with a positive 2. So c is equal to positive 2. So we've solved one
of the unknowns, the each child on
average drinks two cups. So c is equal to 2. So how can we figure
out what a is? Well now we can take
this information and go back into either one
of these and figure out what a has to be. So let's go back into the
orange one right over here and figure out what a has to be. So we had 500a plus 200c, but
we know what c is, c is 2. So 200 times 2 is
equal to 2,900. And now we just
have to solve for a, one equation with one unknown. So we have 500a,
200 times 2 is 400, plus 400 is equal to 2,900. We can subtract 400 from
both sides of this equation. Let me do that. Subtracting 400, and we are
left with this cancels out. And on the left hand
side, we have 500a. This is very exciting. We're in the home stretch. On the right hand
side, you have 2,500. 500a equals 2,500. We can divide both
sides by 500, and we are left with 2,500
divided by 500 is just 5. So you have a is equal
to 5 and you're done. You have solved the
King's conundrum. Each child on average drinks
2 cups of water-- sorry, not cups of water. I don't know where
I got that from. Each child 2 cupcakes and each
adult will eat 5 cupcakes. a is equal to 5. And so based on how many
adults and children are coming to the brunch in
your honor, you now know exactly the
number of cupcakes that the King needs to order.