Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far we have studied linear equations which describe lines in the Cartesian plane.
In the next few lectures, we will explore several types of geometric relationships
that can exist between two linear equations.
One possibility is that the lines described by two linear equations can be parallel.
As we have seen, any non-vertical line in the Cartesian plane
can be uniquely identified by specifying its slope
and a point which lies on the line.
If the point falls on the y-axis
the equation of the line can be written in slope-intercept form.
In this form, the constant m is the slope of the line
and the constant b is the y-coordinate of the y-intercept.
If we write a second equation in slope-intercept form
which uses the same value for m
but a different y-intercept
both lines described by these equations will have the same slope
but no points in common.
We say that these two lines are "parallel" to each other.
Parallel lines have identical slopes
but no points in common.
As a result, parallel lines on a flat plane never intersect.
Let's say that we are given the equation for a line
and are asked to write a second equation for a parallel line
which runs through a given point which does not lie on the y-axis.
Since the two lines are parallel
we know that the slope of the second line must be the same as the slope of the first line.
However, we cannot write the equation using slope-intercept form
since we don't know the second line's y-intercept.
However, we can use point-slope form
since we know the line's slope
and the coordinates x-one and y-one of the given point which lies on the line.
It is easy to tell if two equations written in slope-intercept form describe parallel lines.
If the slope m is the same in both equations
and b is different
then the equations describe two lines with the same slope
but which intercept the y-axis at different points
in other words, two parallel lines.
On the other hand, two equations written in point-slope form
which have the same value for the slope m
may describe parallel lines or they may both describe the same line
even when the two equations contain different values for x-one and y-one.
This is because the point represented by x-one and y-one in the first equation
and the point represented by x-one and y-one in the second equation
may fall on the same line.
In that case, even though both equations look different
they actually describe the same line.
Since in point-slope form, any one of an infinite number of points on the line
may be used to determine the values of x-one and y-one
there are an infinite number of different equations which can be written
that describe the same line.
So when two equations written in point-slope form have the same value for m
but different values for x-one and y-one
how can we determine if they describe parallel lines or the same line?
One easy way to test whether two equations with the same slope describe different lines
is to find where each line crosses the y-axis.
If the two equations intersect the y-axis at different points
then they do not describe the same line.
On the other hand, if two equations with the same slope intersect the y-axis at the same point
then both equations must describe the same line.
The y-intercept for each equation can be found by setting x to zero.
The resulting value of y in each equation is the y-coordinate of each line's y-intercept.
To solve the equations for y we first eliminate the zeros
and complete the arithmetic, multiplying one-half times negative two
to get negative one
and one-half times negative six
to get negative three.
We then add six to both sides of the first equation
and eight to both sides of the second equation
allowing us to cancel the negative and positive six's
and the negative and positive eights.
Completing the arithmetic, negative one plus six is five
and negative three plus eight is also five.
So for both equations, when x is zero, y is five.
Therefore, both equations have the same y-intercept.
Since the equations have the same slope and y-intercept, they both describe the same line.
Now let's try this one more time
but with different values for x-one and y-one in the second equation.
Once again, we find the y-intercepts for each equation
by setting x to zero and solving for y.
We first eliminate the zeros
and complete the arithmetic, multiplying one-half times negative two
to get negative one
and one-half times negative ten
to get negative five.
To rearrange the equations so that y is alone on the left side
we add six to both sides of the first equation
and seven to both sides of the second equation
allowing us to cancel the negative and positive six's
and the negative and positive sevens.
Completing the arithmetic, negative one plus six is five
and negative five plus seven is two.
So when x is zero the value of y is different in each equation.
Each equation has a different y-intercept.
Since both equations have the same slope but different y-intercepts
they describe parallel lines.
In this lecture, we have seen how linear equations can describe parallel lines.
In the next lecture, we will see how linear equations can be created
which describe perpendicular lines.