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# Systems of equations with elimination: apples and oranges

Video transcript

You've gone to a fruit stand
to get some fresh produce. You notice that the
person in front of you gets 5 apples and
4 oranges for $10. You get 5 apples and
5 oranges for $11. Can we solve for the price
of an apple and an orange using this information
in a system of linear equations
in two variables? If yes, what is the solution? If no, what is the
reason we cannot? So we're trying to figure
out the price of an apple and the price of an orange. So I would use a for apple, but
I don't like using o for orange because o looks too
much like a zero. So I'll just say x for apples. Let's let x equal
the price of apples. And let's let y equal
the price of oranges. So let's describe what
happened to the person in line in front of us. They bought 5 apples. So how much did they
spend on apples? Well, they bought 5 apples
times x dollars per apple, so they spent 5x dollars
on their 5 apples. And they bought 4 oranges. They bought 4 oranges
times y dollars per orange. So they spent 4y on oranges. So the total amount that
they spent is 5x plus 4y. And they tell us
that this is $10. This is equal to $10. Now, you get in line,
and you buy 5 apples. So you buy 5 apples, just
like the guy in front of you. And you paid x
dollars per apple. So you're going to pay 5 apples
times the price per apple. This is the amount that
you spend on apples. And then you buy 5 oranges. So you're going to pay 5 oranges
times the price per orange, which is y. So this is how much
you spend on oranges. This is how much you spend on
apples and oranges, the sum. And they tell us
that this is $11. So can we solve
for an x and a y? And it looks like we can. And a big giveaway
right over here is the ratio between the x's and
the y's in these two equations are different. So we're getting some
information here. If the ratios were
exactly the same, if this was 5x plus
4y right over here, and we got a different number,
then we would be in trouble. Because we bought
the same combination, but we got a different price. But the good thing is is that
we have a different combination here. So let's see if we
can work it out. Now, the most obvious
thing that jumps out at me is that I have a 5x here, and
I have a 5x right over here. So if I could subtract
this 5x from that 5x, then I would cancel
out all of the x terms. So what I'm going
to do is I'm going to multiply this bottom
equation by negative 1. So it becomes negative
5x plus negative 5y is equal to negative 11. And then I'm going
to essentially add both of these equations. And I could do that
because I'm doing the same thing to both sides. I already know that this
thing is equal to this thing. So I'm just adding those
things to either side. So on the left hand
side, I have 5x minus 5x. Well, those cancel out. And then I have 4y minus 5y. Well, that's negative y. And that's going to be
equal to 10 minus 11, which is negative 1. And then if we multiply
both sides of this times negative 1, or divide
both sides by negative 1, we're going to get
y is equal to 1. So just like that, we
were able to figure out the price of oranges. It's $1 per orange. So this is equal to 1. Now let's figure out
the price of apples. So we can go back into
either 1 of these equations. I'll go back into this first 1. So 5 times-- so let's go to the
person in line in front of us. They bought 5 apples
at x dollars per apple, plus 4 oranges at $1 per orange,
and they spent a total of $10. So this of course is just 4. Let's subtract 4
from both sides, and we get-- well, 4 times 1
minus 4, that just cancels out. We're just going to be left
with 5x on the left hand side. And on the right hand
side, we have 10 minus 4, which is equal to 6. And we can just divide both
sides by 6 now in order to solve for x. Oh, sorry. We can divide both sides by
5 in order to solve for x. It's late in the day. Brain isn't working. Dividing by 6 wouldn't
have done anything. We would have gotten 5/6x. We just want to get an x here. So dividing both sides by 5, we
get x is equal to 6/5 dollars. Or you could say that
x is equal to 6/5, which is the same
thing as 1 and 1/5, which is the same
thing as $1.20. So it's $1 per orange,
and $1.20 per apple. So we absolutely
could figure out the prices of apples and oranges
using the information given.