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CCSS Math: HSA.REI.D.10, HSA.REI.D.11

You are solving a system
of two linear equations in two variables. You have found more
than one solution that satisfies the system. Which of the following
statements is true? So before even reading
these statements, let's just think
about what's going on. So let me draw my axes here. Let's draw my axes. So this is going to
be my vertical axis. That could be one
of the variables. And then this is
my horizontal axis. That's one of the
other variables. And maybe, for sake of
convention, this could be x, and this could be y, but they're
whatever our two variables are. So it's a system of
two linear equations. So if we're graphing them,
each of the linear equations in two variables can be
represented by a line. Now, there's only
three scenarios here. One scenario is where the
lines don't intersect at all. So the only way
that you're going to have two lines in two
dimensions that don't intersect is if they have the
same slope and they have different y-intercepts. So that's one
scenario, but that's not the scenario that's
being described here. They say, you have found
more than one solution that satisfies the system. Here there are no solutions. So that's not the scenario
that we're talking about. There's another
scenario where they intersect in exactly one place. So they intersect in
exactly one place. There's one point, one
xy-coordinate right over there that satisfies both
of these constraints, but this also is not the
scenario they're talking about. They're telling us that you have
found more than one solution that satisfies the system. So this isn't the
scenario either. So the only other scenario
that we can have-- we don't have parallel lines
that don't intersect. We don't have lines that
only intersect in one place. The only other scenario is that
we're dealing with a situation where both linear
equations are essentially the same constraint. They both are
essentially representing the same xy-relationship. That's the only way that
I can have two lines, and this only applies to
linear relationship and lines. But the only way that
two lines can intersect more than one place is if
they intersect everywhere. So in this situation,
we know that we must have an infinite
number of solutions. So which of these
choices say that? This one right here-- "there are
infinitely many more solutions to the system"--
right over there.