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# Equation with the variable in the denominator

Video transcript

So I have the equation 7 minus
10/x is equal to 2 plus 15/x. And so this isn't
the type of equation that you might think that
you're used to solving. But I'll give you a
few moments to see if you can solve it on your own. Well, what we'll see is we
can do a quick multiplication of both sides to actually
simplify this to a form that we are more
used to looking at. So what's probably bothering
you, because it's bothering me, is these x's that we have in the
denominators right over here. We're like, well, how
do we deal with that? Well, whenever we see
an x in the denominator, the temptation is
to multiply it by x. But we can't just multiply
one of the terms by x. We have to multiply
the entire side by x. So we could multiply
this entire side by x. But we can't just multiply
the left-hand side by x. We'd also want to multiply
the right-hand side by x. And so what will that give us? Well, we distribute the x. We get x times 7 is 7x. And then x times
negative 10/x, well, that's just going
to be negative 10. So you get negative
10 right over there. So the left-hand side
simplifies to 7x minus 10. And then your right-hand
side, once again, distribute the x.
x times 2 is 2x. x times 15/x, well,
x times something over x is just going to be
the something. x times 15/x is just going to
be 15-- plus 15. So now we've simplified
this to a linear equation. We have the variable
on both sides. So we just have to do
some of the techniques that we already know. So the first thing
that I like to do is maybe get all my x's
on the left-hand side. So I want to get rid of
this 2x right over here. So I subtract 2x from
the right-hand side. Now, and I always remind
you, I can't do that just to the right-hand side. If I did it just to
the right-hand side, it wouldn't be an
equality anymore. You have to do that to the
left-hand side as well. And so we are left with-- let
me get that pink color again. On the left-hand side, 7x, 7 of
something minus 2 of something, well, you're going to have 5
of that something, minus 10. These two x's negate each other. And you're left with equals 15. Now we can get rid
of this negative 10 by adding 10 to both sides. You know, I like
that green color when I do stuff to both sides. So I can add 10 to both sides. And I'm left with
5x-- these negate each other-- is equal to 25. And this is the home stretch. You see where this is going. We can divide both sides by 5. And we are left with
x is equal to 5. Now let's verify that
this actually worked. So let's go back to
the original equation. We have 7 minus 10/5. This needs to be
equal to-- I'm just taking our 5 and
substituting it back here. This needs to be
equal to 2 plus 15/5. So this is 7 minus 10/5. This is just 2. It needs to be equal to 2
plus 15/5, which is just 3. So 2 plus 3, 7 minus
2 is 5, 2 plus 3 is 5, 5 is indeed equal to 5. And we are done.