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# Solutions to systems of equations: dependent vs. independent

A dependent system of equations has infinite solutions, and an independent system has a single solution. Watch an example of analyzing a system to see if it's dependent or independent. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Is the system of
linear equations below dependent or independent? And they give us two
equations right here. And before I tackle
this specific problem, let's just do a little
bit a review of what dependent or independent means. And actually, I'll compare that
to consistent and inconsistent. So just to start
off with, if we're dealing with systems of linear
equations in two dimensions, there's only three possibilities
that the lines or the equations can have relative to each other. So let me draw the
three possibilities. So let me draw three
coordinate axes. So that's my first
x-axis and y-axis. Let me draw another one. That is x and that is y. Let me draw one
more, because there's only three possibilities
in two dimensions. x and y if we're dealing
with linear equations. So you can have the situation
where the lines just intersect in one point. Let me do this. So you could have
one line like that and maybe the other line
does something like that and they intersect at one point. You could have the
situation where the two lines are parallel. So you could have a situation--
actually let me draw it over here-- where you have one
line that goes like that and the other line has the
same slope but it's shifted. It has a different y-intercept,
so maybe it looks like this. And you have no points
of intersection. And then you could
have the situation where they're actually the same
line, so that both lines have the same slope and
the same y-intercept. So really they
are the same line. They intersect on an
infinite number of points. Every point on
either of those lines is also a point
on the other line. So just to give you a little
bit of the terminology here, and we learned this in the
last video, this type of system where they don't intersect,
where you have no solutions, this is an inconsistent system. And by definition, or
I guess just taking the opposite of
inconsistent, both of these would be considered consistent. But then within consistent,
there's obviously a difference. Here we only have one solution. These are two different lines
that intersect in one place. And here they're essentially
the same exact line. And so we differentiate
between these two scenarios by calling this one
over here independent and this one over
here dependent. So independent-- both lines
are doing their own thing. They're not dependent
on each other. They're not the same line. They will intersect
at one place. Dependent-- they're
the exact same line. Any point that satisfies one
line will satisfy the other. Any points that
satisfies one equation will satisfy the other. So with that said, let's see if
this system of linear equations right here is dependent
or independent. So they're kind of
having us assume that it's going
to be consistent, that we're going to
intersect in one place or going to intersect in an
infinite number of places. And the easiest way to
do this-- we already have this second equation here. It's already in
slope-intercept form. We know the slope is negative
2, the y-intercept is 8. Let's put this first equation
up here in slope-intercept form and see if it has a different
slope or a different intercept. Or maybe it's the same line. So we have 4x plus
2y is equal to 16. We can subtract 4x
from both sides. What we want to do is isolate
the y on the left hand side. So let's subtract
4x from both sides. The left hand side-- we
are just left with a 2y. And then the right hand side,
we have a negative 4x plus 16. I just wrote the negative
4 in front of the 16, just so that we have it in the
traditional slope-intercept form. And now we can divide both
sides of this equation by 2, so that we can isolate
the y on the left hand side. Divide both sides by 2. We are left with y is equal
to negative 4 divided by 2 is negative 2x plus
16 over 2 plus 8. So all I did is algebraically
manipulate this top equation up here. And when I did that, when
I solved essentially for y, I got this right
over here, which is the exact same thing
as the second equation. We have the exact same slope,
negative 2, negative 2, and we have the exact
same y-intercept, 8 and 8. If I were to graph these
equations-- that's my x-axis, and that is my y-axis-- both
of them have a y-intercept at 8 and then have a
slope of negative 2. So they look
something-- I'm just drawing an approximation of it--
but they would look something like that. So maybe this is the graph
of this equation right here, this first equation. And then the second equation
will be the exact same graph. It has the exact
same y-intercept and the exact same slope. So clearly these two
lines are dependent. They have an infinite
number of points that are common to both of them,
because they're the same line.