Professor Dave here, let’s graph some lines.
We just learned about the coordinate plane and how to represent equations graphically.
Let’s continue with this topic by finding the equation of a line given some data, and
plotting that line graphically in the form of Y equals MX plus B. As we know, X and Y
are variables, while M represents the slope of the line and B is the Y intercept, or the
Y value for the point where the line crosses the Y axis.
Since any two points define a line, we could list any two points, and find the equation
that describes the line that passes through these points.
Let’s try (negative two, three), and (one, five).
First, we can calculate the slope.
Again, that’s rise over run, which is five minus three over one minus negative two, or
two thirds.
Let’s plug that into our equation, giving us Y equals two thirds X plus B. Now how are
we going to find B?
Well, since this equation relates the X and Y coordinates of any point on this line, we
can simply plug in the coordinates for any point on this line, and we should be able
to solve for B. Let’s use one, five, as our X and Y values.
We plug in one for X and five for Y, and we get five equals two thirds times one plus
B. Let’s change five into fifteen thirds, and then subtract two thirds from both sides.
That leaves us with thirteen thirds.
Just to be thorough, let’s plug in the other point, and verify that we get the same thing.
Three, or nine thirds, equals negative four thirds plus B. Add four thirds to both sides,
and we see that we get the same thing as before.
And with that, we have our equation for this line, Y equals two thirds X plus thirteen
thirds.
So now we know how to find the equation of a line given any two points on that line.
If we are given one point and the slope, things are even easier, because then we just have
to solve for B. Say the line goes through four negative seven and it has a slope of
negative one half.
The equation must be Y equals negative one half X plus B. Then we just plug in the coordinates
for the one point we have, and we get negative seven equals negative one half times four
plus B. Negative seven equals negative two plus B, and B equals negative five.
This allows us to write the entire equation.
If we want to graph any of these lines, it’s just a matter of marking two points on the
line and carefully drawing the line that passes through them, extending infinitely in either
direction, which is what is meant by these little arrows on either side, that they continue
forever.
The two points can be ones that were given, they could be an intercept and another point
given by applying the slope in either direction, or if we just have the equation we could plug
in any value for X to get the corresponding Y value, and therefore an ordered pair representing
a point on the line, and then do the same thing for another X value to get another point.
Whatever we need to do to get two points will be enough to draw the line that represents
the linear equation.
If that’s all clear, let’s check comprehension.