Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far we have studied two different forms of linear equations
which describe lines in the Cartesian plane.
Either form may be more useful
depending upon what information is known about a line's graph.
If we know the line's slope and y-intercept
then the "slope-intercept" form is the simplest way to write an equation
which describes the line.
The slope-intercept form is written as "y = mx + b"
where the constant "m" is the line's slope
and the constant "b" is the y-coordinate of the point where the line crosses the y-axis.
On the other hand, if we don't know the y-intercept
but we do know the coordinates of some other point which lies on the line
then the "point-slope" form is usually a more convenient way to write the line's equation.
The point-slope form is written as
"y minus y-one" equals "m" times the quantity "x minus x-one"
where "m" is the line's slope
and the constants x-one and y-one
are the coordinates of a known point which lies on the line.
In this chapter, we will introduce a third form of equation for a line
called the "two-point" form.
We know from Geometry that we cannot uniquely describe a line, knowing only its slope
since any plane contains an infinite number of parallel lines with the same slope.
Likewise, we cannot uniquely describe a line by specifying a point that lies on the line
without specifying the slope
since there are an infinite number of lines with different slopes
which include any given point.
To uniquely describe a line on a plane
we must specify both its slope
and at least one point which the line includes.
Both the slope-intercept and the point-slope forms are based on this type of description.
The difference is that the slope-intercept form requires that the specified point
lie on the y-axis
while the point-slope form allows the given point to be anywhere on the xy-plane.
But there is another way to describe a line without specifying its slope.
A line can be uniquely described by specifying two distinct points which the line includes.
This principal is the basis for the "two-point" form of the equation for a line.
This form does not require a line's slope to be known
in order to create the line's equation.
All that is required are the coordinates of two points which lie on the line.
The two-point form can be created by starting with the point-slope form
and replacing the line's slope "m"
with the formula for slope which we learned in the chapter on slope.
As you may recall, the formula for slope uses the coordinates of two distinct points
through which the line passes.
The coordinates of one point x-one and y-one
are constants already contained in the point-slope equation.
But the coordinates of the second point x-two and y-two
are additional constants which are required by the two-point form.
To demonstrate how to use the two-point form
let's try an example, using this form to write the equation for a line.
Let's say that we are given the graph of a line, but not its equation.
All we know about the line is that it passes through the points (3,5) and (7,8).
Using the two-point form of the equation for a line
we assign the coordinates of the first point, three and five
to the constants x-one and y-one.
Since x-one is three and y-one is five
we replace x-one in the equation with three
and y-one with five.
Likewise, we assign the coordinates of the second point, seven and eight
to the constants x-two and y-two.
And so in the equation, x-two and y-two
are replaced with seven and eight.
Completing the arithmetic
eight minus five is three
and seven minus three is four.
Of course, we could have accomplished the same thing
by first using the slope formula
to calculate the slope from the coordinates of the two known points
and then substituting this value for slope into the point-slope form
along with the coordinates of either one of the points.
Like the slope-intercept and point-slope forms
the two-point form cannot be used to describe a vertical line.
This is because the x-coordinates of all points on a vertical line have the same value.
No matter which two points we pick for the two-point form
the constants x-one and x-two will be equal.
Since x-one and x-two have the same value
the denominator x-two minus x-one in the equation
will be zero.
The equation will therefore include a term divided by zero, which is undefined.
There are many different forms of equations which describe lines in the xy-plane.
So far we have discussed three of the most common forms.
Any of these forms may be used to describe a specific line.
The most convenient form to use depends upon what information is known about the line.
In the next lecture, we will introduce one additional form of linear equation
called the "standard form".