Main content

## Number of solutions to systems of equations

Current time:0:00Total duration:6:36

# Number of solutions to a system of equations graphically

CCSS Math: 8.EE.C.8, 8.EE.C.8a, HSA.REI.D.10, HSA.REI.D.11

## Video transcript

So that we don't get stumped
by the Arbeglas in our life and especially when we don't
have talking birds to help us, we should be able to
identify when things get a little bit weird with
our systems of equations. When we have scenarios that have
an infinite number of solutions or that have no
solutions at all. And just as a little bit of a
review of what could happen, these are the-- think
about the three scenarios. You have the first
scenario which is kind of where we started off,
where you have two systems that just intersect in one place. And then you have
essentially one solution. So if you were to
graphically represent it you have one solution right
over there, one solution. And this means that the two
constraints are consistent and the two constraints are
independent of each other. They're not the exact same line,
consistent and independent. Then you have the other scenario
where they're consistent, they intersect,but they're
essentially the same line. They intersect everywhere. So this is one of the
constraints for one of the equations,
and the other one if you look at it,
if you graph it, it is actually the
exact same one. So here you have an infinite
number of solutions. It's consistent, you
do have solutions here, but they're dependent equations. It's a dependent system. And then the last
scenario, and this is when you're dealing
in two dimensions, the last scenario is where
your two constraints just don't intersect with each other. One might look like
this, and then the other might look like this. They have the exact
same slope but they have different intercepts. So this there is no solution,
they never intersect. And we call this an
inconsistent system. And if you wanted to think
about what would happen just think about what's
going on here. Here you have different slopes. And if you think about
it, two different lines with different
slopes are definitely going to intersect
in exactly one place. Here they have the same
slope and same y-intercept, so you have an infinite
number of solutions. Here you have the same slope
but different y-intercepts, and you get no solutions. So the times when
you're solving systems where things are going
to get a little bit weird are when you have
the same slope. And if you think about it,
what defines the slope, and I encourage you to test this
out with different equations, is when you have-- if you
have your x's and y's, or you have your a's and b's or
you have your variables on the same side of
an equation, where they have the same ratio
with respect to each other. So with that, kind of
keeping that in mind, let's see if we can think
about what types of solutions we might find. So let's take this down. So they say determine
how many solutions exist for the
system of equations. So you have 10x minus
2y is equal to 4, and 10x minus 2y is equal to 16. So just based on what we
just talked about the x's and the y's are on the
same side of the equation and the ratio is
10 to negative 2. Same ratio. So something strange is
going to happen here. But when we have the
same kind of combination of x's and y's in the first one
we get 4, and on the second one we get 16. So that seems a
little bit bizarre. Another way to
think about it, we have the same number of
x's, the same number of y's but we got a different number
on the right hand side. So if you were to simplify
this, and we could even look at the hints
to see what it says, you'll see that
you're going to end up with the same slope but
different y-intercepts. So we convert both the slope
intercept form right over here and you see one, the blue one
is y is equal to 5x minus 2, and the green one is y
is equal to 5x minus 8. Same slope, same ratio
between the x's and the Y's, but you have different
values right over here. You have different y-intercepts. So here you have no solutions. That is this scenario right over
here if you were to graph it. So no solutions,
check our answer. Let's go to the next question. So let's look at this
one right over here. So we have negative 5 times
x and negative 1 times y. We have 4 times x and 1 times y. So it looks like the
ratio if then we're looking at the x's and y's
always on the left hand side right over here, it looks like
the ratios of x's and y's are different. You have essentially
5 x's for every one y, or you could say negative 5
x's for every negative 1 y, and here you have 4
x's for every 1 y. So this is fundamentally
a different ratio. So right off the bat
you could say well these are going to intersect
in exactly one place. If you were to put this
into slope intercept form, you will see that they
have different slopes. So you could say
this has one solution and you can check your answer. And you could look at the
solution just to verify. And I encourage you to do this. So you see the blue one if you
put in the slope intercept form negative 5x plus 10 and
you take the green one into slope intercept
form negative 4x minus 8. So different slopes,
they're definitely going to intersect
in exactly one place. You're going to
have one solution. Let's try another one. So here we have 2x plus
y is equal to negative 3. And this is pretty
clear, you have 2x plus y is equal
to negative 3. These are the exact
same equations. So it's consistent information,
there's definitely solutions. But there's an infinite number
of solutions right over here. This is a dependent system. So there are infinite
number of solutions here and we can check our answer. Let's do one more because that
was a little bit too easy. OK so this is interesting
right over here, we have it in different forms. 2x plus y is equal
to negative 4, y is equal to
negative 2x minus 4. So let's take this
first blue equation and put it into
slope intercept form. If we did that you would
get, if you just subtract 2x from both sides you get y is
equal to negative 2x minus 4, which is the exact same thing as
this equation right over here. So once again they're
the exact same equation. You have an infinite
number of solutions. Check our answer, and you can
look at the solution right here. You convert the blue
one into slope intercept and you get the exact
same thing as what you saw in the green one.