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## Analyzing the number of solutions to linear equations

## Video transcript

Determine the
number of solutions for each of these
equations, and they give us three equations right over here. And before I deal with these
equations in particular, let's just remind
ourselves about when we might have one or
infinite or no solutions. You're going to
have one solution if you can, by
solving the equation, come up with something like
x is equal to some number. Let's say x is
equal to-- if I want to say the abstract--
x is equal to a. Or if we actually
were to solve it, we'd get something like x
equals 5 or 10 or negative pi-- whatever it might be. But if you could actually
solve for a specific x, then you have one solution. So this is one solution,
just like that. Now if you go and you try to
manipulate these equations in completely legitimate
ways, but you end up with something crazy
like 3 equals 5, then you have no solutions. And if you just think
about it reasonably, all of these equations
are about finding an x that satisfies this. And if you were to just
keep simplifying it, and you were to get
something like 3 equals 5, and you were to ask
yourself the question is there any x that can somehow
magically make 3 equal 5, no. No x can magically
make 3 equal 5, so there's no way that you could
make this thing be actually true, no matter
which x you pick. So if you get something
very strange like this, this means there's no solution. On the other hand, if you get
something like 5 equals 5-- and I'm just over
using the number 5. It didn't have to
be the number 5. It could be 7 or 10
or 113, whatever. And actually let
me just not use 5, just to make sure that you
don't think it's only for 5. If I just get something,
that something is equal to itself,
which is just going to be true no matter what
x you pick, any x you pick, this would be true for. Well, then you have
an infinite solutions. So with that as a
little bit of a primer, let's try to tackle
these three equations. So over here, let's see. Maybe we could subtract. If we want to get rid of this
2 here on the left hand side, we could subtract
2 from both sides. If we subtract 2
from both sides, we are going to be left
with-- on the left hand side we're going to be
left with negative 7x. And on the right
hand side, you're going to be left with 2x. This is going to
cancel minus 9x. 2x minus 9x, If we simplify
that, that's negative 7x. You get negative 7x is
equal to negative 7x. And you probably see
where this is going. This is already true
for any x that you pick. Negative 7 times that x is going
to be equal to negative 7 times that x. So we already are going
into this scenario. But you're like hey, so
I don't see 13 equals 13. Well, what if you did
something like you divide both sides by negative 7. At this point, what I'm
doing is kind of unnecessary. You already understand that
negative 7 times some number is always going to be
negative 7 times that number. But if we were to do this,
we would get x is equal to x, and then we could subtract
x from both sides. And then you would
get zero equals zero, which is true for
any x that you pick. Zero is always going
to be equal to zero. So any of these
statements are going to be true for any x you pick. So for this equation
right over here, we have an infinite
number of solutions. Let's think about this one
right over here in the middle. So once again, let's try it. I'll do it a little
bit different. I'll add this 2x and this
negative 9x right over there. So we will get negative 7x
plus 3 is equal to negative 7x. So 2x plus 9x is
negative 7x plus 2. Well, let's add-- why don't we
do that in that green color. Let's do that in
that green color. Plus 2, this is 2. Now let's add 7x to both sides. Well if you add 7x to
the left hand side, you're just going to
be left with a 3 there. And if you add 7x to
the right hand side, this is going to go
away and you're just going to be left with a 2 there. So all I did is I added 7x. I added 7x to both
sides of that equation. And now we've got
something nonsensical. I don't care what x you pick,
how magical that x might be. There's no way that that x is
going to make 3 equal to 2. So in this scenario right over
here, we have no solutions. There's no x in the universe
that can satisfy this equation. Now let's try this
third scenario. So once again, maybe we'll
subtract 3 from both sides, just to get rid of
this constant term. So we're going to get negative
7x on the left hand side. On the right hand side, we're
going to have 2x minus 1. And now we can subtract
2x from both sides. To subtract 2x from
both sides, you're going to get-- so
subtracting 2x, you're going to get negative
9x is equal to negative 1. Now you can divide both
sides by negative 9. And you are left with
x is equal to 1/9. So we're in this
scenario right over here. We very explicitly
were able to find an x, x equals 1/9, that
satisfies this equation. So this right over here
has exactly one solution.