 
In this lecture,
we're going to look
at how solutions to equations
and inequalities in two
variables look like.
So remember in the
last module, when
we solved equations
or inequalities in one
variable, the solution
could be represented either
as points on a number line
or infinitely many points
as intervals on a number line.
So when we are
plotting solutions
to equations and inequalities
in two variables,
the result is either
a curve or a region
bounded by these
curves in the xy-plane.
In this module, we'll
investigate equations
that represent lines, circles,
and a few other types of curves
and specific regions that
are bounded by these curves.
The idea of describing geometric
curves with algebraic equations
is called analytic geometry.
I can tell you for
myself, as a student,
I really was drawn to
mathematics precisely because
of analytic geometry.
I was so amazed
that equations could
be thought of as
these beautiful curves
that you are about to see.
 
All right, so let's
take a look at how
we can play with some
of these equations.
A crude tool that
is available to you
right now in order to figure
out how something looks like
is to plot as many solution
points to the equation
as you need.
So you can see a pattern emerge.
There are many other
sophisticated tools
that you'll see in
college algebra, calculus,
that give you other
methods of curve sketching.
So let's start with testing
points in the xy-plane
to see if there are solutions to
a given equation or inequality
or not.
So in order for our point in an
xy-plane, a comma b-- remember,
this is the x-coordinate,
this is the y-coordinate--
to be a solution to an equation
or an inequality is if we
replace a for x and b for y
in the equation and you end up
with a true statement, then the
point a,b is a solution to that
equation or inequality.
Let's take some
concrete examples.
So we are asking you, are
these points, the four points
listed here, solutions to this
equation, y equals x minus 1?
So let's make a table.
Because that will make
it easy for us to see.
So we have our equation,
which is y equals x minus 1.
Our point, let's start with
the first point, negative 2, 1.
This negative 2 is the
value of x-coordinate.
1 is the value of y-coordinate.
So we substitute that
in this equation.
And we'll end up with 1
equals negative 2 minus 1 or 1
equals negative 3, which
is a false statement.
So false statement, what that
means is that negative 2 comma
1 is not a solution
to this equation.
If it were, you should
get a true statement.
All right, pause the video here.
And do this next
point on your own.
And then we'll discuss it.
Go ahead, pause it.
 
All right, so you replace
x equals 2 and y equals 1.
And look what we get.
We get 1 equals 1, which
is a true statement.
And that means 2, 1 is a
solution to our equation y
equals x minus 1.
All right, try the next
two points on your own.
Pause the video.
And then we'll discuss it.
 
Assuming you've
paused the video.
And you've come back.
So let's do the next two.
So x is negative 5 and y is 0.
Plug it in.
And you get 0 equals negative
6, which is a false statement.
So negative 5, 0
is not a solution.
0, negative 1, negative
1 equals 0 minus 1.
Negative 1 equals negative 1.
So true.
So 0, negative 1 is a
solution to the equation y
equals x minus 1.
So it's relatively easy to
check whether a given point is
a solution to this
equation or not.
OK, let's try inequality.
Again, same thing, you're
going to make a chart.
Substitute the point
negative 2 for x, 1 for y.
And we get 1 is greater
than negative 5, which
is a true statement.
So negative 2, 1 is a
solution to y greater than 2x
minus 1 inequality.
However, look.
So is any point
negative 2y, as long
as the y would be
greater than negative 5.
Because if you put a y
here and you end up with y,
y, y, y, greater
than negative 5.
So as long as your y-coordinate
is bigger than negative 5,
could be 1, could
be negative 1, 100.
So negative 2 comma
any y-coordinate
that is bigger
than negative 5 is
going to be a solution
to this inequality.
So there are infinitely many
solutions to this inequality.
All right, you try
the points 2, 3,
negative 5, 0,
and 4, negative 1.
And see if they are
solutions to this inequality.
Pause the video here, please.
Assuming you have
paused the video.
And you've come back.
I'll share the
solutions with you here.
All right, so 2, 3.
You plug it in.
You get 3 is bigger
than 3, which is false.
So 2, 3 is not a solution.
When you plug in negative 5,
0, 0 greater than negative 11
is a true statement.
So negative 5, 0 is a solution.
And in fact, as long as you
use any y-coordinate bigger
than negative 11,
negative 5, y, where
y is better than negative
1, are all solutions
to this inequality.
So negative 5, negative
10, negative 9, 100,
any y-coordinate,
as long as x is
negative 5 and y is
bigger than negative 1,
are all solutions
to this inequality.
What about 4, negative 1?
We get negative 1 bigger than
7, which is a false statement.
So 4, negative 1
is not a solution.
All right, let's take a look
at, what if we give you y
equals 2x minus 1
is our equation,
and we only give you
one of the coordinates?
Can you find the
missing coordinate?
So let's try these problems
and see what you can do.
Again, make a chart
similar to the one
we were just looking at.
And that will help you.
So pause the video.
And try on your own first.
 
All right, so we
have negative 2,
y. x is negative 2 and
y-coordinate is y-coordinate.
So you plug it in.
x equals negative 2.
Solve for y.
And you get y equals negative 5.
What that means is that if you
want x to be the negative 2--
x is fixed, x is negative
2-- then y-coordinate
will have to be negative 5.
So negative 2, negative 5 is
a solution to this equation.
 
If you wanted x
comma 3, which means,
what x-coordinate can we have
so that when y-coordinate is 3,
this point is a solution
to the equation?
So you plug in x.
You plug in 3 for y.
Solve for x.
And look what we get.
We get x equals 2.
So 2 comma 3 is a solution.
Again, if you do 0, y, so
you're replacing x to 0
and solving for y, you end
up with y equals negative 1.
Such a coordinate,
when x-coordinate is 0,
is called the y-intercept.
Because if you were to plot
it, it will be on the y-axis.
When the y-coordinate
is 0, this point
is called the x-intercept.
Because when you
plot it, this point
is going to be on the x-axis.
So plug in y equals 0.
Solve for x.
You get x equals 1/2.
So that's how you find
missing coordinates.
We are doing this
exercise because that's
how you decide what points
are solutions to an equation.
You pick any coordinate.
Solve for the
remaining variable.
And you are now
ready to plot points.
You can do the same
for inequalities.
So, again, let's
do the same thing.
So you can solve for y.
And you get y greater than 3.
That means 2 comma, y,
where y is bigger than 3,
are all solutions
to this inequality.
So 4, 3.1, 5.1, could be
2 comma any y-coordinate,
as long as it's bigger than
3, you have a solution.
Same process when you have
y-coordinate is negative 1
and you want to find x.
So 0 greater than
x, which means x
has to be smaller
than 0 when you solve.
So any coordinate for which
x-coordinate is smaller than 0
is going to work with you, as
long as the y-coordinate is
negative 1.
So negative 4,
negative 1, negative 1,
negative 1, and so on.
 
All right, so what happens here?
Then say, y is bigger
than negative 1.
So 0 is the x-coordinate, fixed.
y-coordinate has to
be any coordinate,
as long as it's bigger
than negative 1.
So 0, 1, 2, 1.1,
4.5, 5, all of those
will form solutions
to this inequality.
Let's do x, 0.
Again, replace y to 0.
Solve for x.
So x is smaller than 1/2.
So you can have 0.
You can have
negative 1 and so on.
So these are all solutions
to the inequality
when y-coordinate is 0.
 
