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## Systems of equations word problems

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# Systems of equations with elimination: TV & DVD

CCSS.Math: , , , ,

## Video transcript

An electronics warehouse ships
televisions and DVD players in certain combinations
to retailers throughout the country. They tell us that the weight
of 3 televisions and 5 DVD players is 62.5 pounds, and the
weight of 3 televisions and 2 DVD players-- so they're
giving us different combinations-- is 52 pounds. Create a system of
equations that represents this situation. Then solve it to find out how
much each television and DVD player weighs. Well, the two pieces of
information they gave us in each of these statements can be
converted into an equation. The first one is is that the
weight of 3 televisions and 5 DVD players is 62.5 pounds. Then they told us that the
weight of 3 televisions and 2 DVD players is 52 pounds. So we can translate these
directly into equations. If we let t to be the weight of
a television, and d to be the weight of a DVD player, this
first statement up here says that 3 times the weight
of a television, or 3 televisions, plus 5 times the
weight of a DVD player, is going to be equal
to 62.5 pounds. That's exactly what this first
statement is telling us. The second statement, the weight
of 3 televisions and 2 DVD players, so if I have 3
televisions and 2 DVD players, so the weight of 3 televisions
plus the weight of 2 DVD players, they're telling us
that that is 52 pounds. And so now we've set up the
system of equations. We've done the first part,
to create a system that represents the situation. Now we need to solve it. Now, one thing that's especially
tempting when you have two systems, and both of
them have something where, you know, you have a 3t here and
you have a 3t here, what we can do is we can multiply one of
the systems by some factor, so that if we were to add this
equation to that equation, we would get one of the terms
to cancel out. And that's what we're going
to do right here. And you can do this, you can
do this business of adding equations to each other, because
remember, when we learned this at the beginning of
algebra, anything you do to one side of an equation, if
I add 5 to one side of an equation, I have to add 5 to
another side of the equation. So if I add this business to
this side of the equation, if I add this blue stuff to the
left side of the equation, I can add this 52 to the
right-hand side, because this is saying that 52 is the same
thing as this thing over here. This thing is also 52. So if we're adding this to the
left-hand side, we're actually adding 52 to it. We're just writing it
a different way. Now, before we do that, what I
want to do is multiply the second, blue equation
by negative 1. And I want to multiply
it by negative 1. So negative 3t plus-- I could
write negative 2d is equal to negative 52. So I haven't changed the
information in this equation. I just multiplied everything
by negative 1. The reason why I did that is
because now if I add these two equations, these 3t terms
are going to cancel out. So let's do that. Let's add these two equations. And remember, all we're doing is
we're adding the same thing to both sides of this
top equation. We're adding essentially
negative 52 now, now that we've multiplied everything
by a negative 1. This negative 3t plus negative
2d is the same thing as negative 52. So let's add this left-hand
side over here. The 3t and the negative
3t will cancel out. That was the whole point. 5d plus negative 2d is 3d. So you have a 3d is equal to
62.5 plus negative 52, or 62.5 minus 52 is 10.5. And now we can divide both sides
of this equation by 3. And you get d is equal
to 10.5 divided by 3. So let's figure out
what that is. 3 goes into 10.5-- it goes
into 10 three times. 3 times 3 is 9. Subtract. Get 1. Bring down the 5. Of course, you have your decimal
point right here. 3 goes into 15 five times. 5 times 3 is 15. You've got to subtract,
and you get a 0. So it goes exactly 3.5 times. So the weight of a DVD player--
that's what d represents-- is 3.5 pounds. Now we can substitute back into
one of these equations up here to figure out the weight
of a television. We can just use that
top equation. So you get 3t plus 5 times the
weight of a DVD player, which we just figured out is 3.5. Remember, we're just looking for
values that satisfy both of these equations. So 5 times 3.5-- needs
to be equal to 62.5. So you get 3t plus-- what
is this going to be? This is going to be 15
plus 2.5, right? 5 times 0.5 is 2.5,
5 times 3 is 15. So it's 17.5, is
equal to 62.5. Now we can subtract 17.5 from
both sides of this equation. And what do we get? The left-hand side is
just going to be 3t. This cancels out, that was
the whole point of it. 3t is going to be equal
to-- let's see. The 0.5 minus 0.5,
that cancels out. So this is the same thing
as 62 minus 17. 62 minus 7 would be 55. And so we're going to
subtract another 10. So it's going to be 45. So this is going to
be equal to 45. Now you can divide both sides
of this equation by 3. And we get t is equal to 15. So we've solved our system. The weight of a DVD player is
3.5 pounds, and the weight of a television is 15 pounds. And we're done.