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# Solutions to systems of equations: consistent vs. inconsistent

A

*system of equations has at least one solution, and an***consistent***system has no solution. Watch an example of analyzing a system to see if it's consistent or inconsistent. Created by Sal Khan and Monterey Institute for Technology and Education.***inconsistent**Video transcript

Is the system of
linear equations below consistent or inconsistent? And they give us x
plus 2y is equal to 13 and 3x minus y is
equal to negative 11. So to answer this
question, we need to know what it means to be
consistent or inconsistent. So a consistent
system of equations. has at least one solution. And an inconsistent system of
equations, as you can imagine, has no solutions. So if we think about
it graphically, what would the graph of a
consistent system look like? Let me just draw a
really rough graph. So that's my x-axis,
and that is my y-axis. So if I have just two
different lines that intersect, that would be consistent. So that's one line, and
then that's another line. They clearly have
that one solution where they both
intersect, so that would be a consistent system. Another consistent
system would be if they're the same
line, because then they would intersect at
a ton of points, actually at an infinite
number of points. So let's say one of the
lines looks like that. And then the other line is
actually the exact same line. So it's exactly
right on top of it. So those two intersect at
every point along those lines, so that also would
be consistent. An inconsistent system
would have no solutions. So let me again draw my axes. Let me once again draw my axes. It will have no solutions. And so the only way
that you're going to have two lines
in two dimensions have no solutions is if
they don't intersect, or if they are parallel. So one line could
look like this. And then the other line
would have the same slope, but it would be shifted over. It would have a
different y-intercept, so it would look like this. So that's what an inconsistent
system would look like. You have parallel lines. This right here is inconsistent. So what we could do is
just do a rough graph of both of these lines
and see if they intersect. Another way to do it is,
you could look at the slope. And if they have the same slope
and different y-intercepts, then you'd also have
an inconsistent system. But let's just graph them. So let me draw my x-axis
and let me draw my y-axis. So this is x and then this is y. And then there's a couple
of ways we could do it. The easiest way is really
just find two points on each of these that satisfy
each of these equations, and that's enough
to define a line. So for this first one, let's
just make a little table of x's and y's. When x is 0, you have
2y is equal to 13, or y is equal to 13/2, which
is the same thing as 6 and 1/2. So when x is 0, y is 6 and 1/2. I'll just put it
right over here. So this is 0 comma 13/2. And then let's just see
what happens when y is 0. When y is 0, then
2 times y is 0. You have x equaling 13. x equals 13. So we have the point 13 comma 0. So this is 0, 6 and
1/2, so 13 comma 0 would be right about there. We're just trying to
approximate-- 13 comma 0. And so this line right
up here, this equation can be represented by this line. Let me try my best to draw it. It would look
something like that. Now let's worry about this one. Let's worry about that one. So once again, let's make a
little table, x's and y's. I'm really just looking for
two points on this graph. So when x is equal to
0, 3 times 0 is just 0. So you get negative y
is equal to negative 11, or you get y is equal to 11. So you have the point 0, 11, so
that's maybe right over there. 0 comma 11 is on that line. And then when y is 0,
you have 3x minus 0 is equal to negative 11, or
3x is equal to negative 11. Or if you divide
both sides by 3, you get x is equal
to negative 11/3. And this is the exact same
thing as negative 3 and 2/3. So when y is 0, you have x
being negative 3 and 2/3. So maybe this is about
6, so negative 3 and 2/3 would be right about here. So this is the point
negative 11/3 comma 0. And so the second equation will
look like something like this. Will look something like that. Now clearly-- and I might have
not been completely precise when I did this hand-drawn
graph-- clearly these two guys intersect. They intersect right over here. And to answer
their question, you don't even have to find the
point that they intersect at. We just have to
see, very clearly, that these two lines intersect. So this is a consistent
system of equations. It has one solution. You just have to have at least
one in order to be consistent. So once again, consistent
system of equations.