Professor Dave again, let’s look at some more lines.
We know how to graph lines in a few different forms, so let’s start comparing different
types of lines to see what we can learn from this.
First, let’s take a look at a graph containing two parallel lines.
These are lines in the same plane that never intersect.
In order for this to be the case, the lines must have the same ratio of vertical change
to horizontal change.
In other words, they must have equivalent slopes.
Their equations will be different, but their slopes are identical.
On the other hand, two lines that intersect at a right angle are perpendicular to one
another.
The relationship between the slopes of perpendicular lines is not as obvious as it was for parallel
lines, but as it turns out, perpendicular lines have slopes that are opposite reciprocals
of one another.
By opposite reciprocal, we mean that if one has a slope of two, the other has a slope
of negative one half.
If one has a slope of negative five, the other has a slope of one fifth.
We flip the term, and flip the sign.
There are many questions you may encounter in algebra that require you to understand
these concepts.
For example, a question may ask you to write the equation of any line that is perpendicular
to Y equals negative one fourth X plus two.
If you are unaware of the relationship between the slopes of perpendicular lines, you might
find yourself graphing this line and then choosing an arbitrary perpendicular line so
you can find its equation.
But instead of all this, we just recognize that the opposite reciprocal of this slope
would be positive four, so any line with a slope of four will satisfy the requirements.
Y equals four X plus six, or four X minus a thousand, it doesn’t matter, if it has
a slope of four, it will intersect this line at a right angle somewhere.
Some questions will require that you be a little more specific.
You could be asked to find the equation of a line that is parallel to Y equals three
X plus one and goes through the point four, four.
Well if it’s parallel, we know it has a slope of three, so Y equals three X plus B.
Now we can plug in the coordinates for the one point we know, so four equals three times
four plus B, and if we solve for B, we get negative eight, so the equation must be Y
equals three X minus eight.
We could do the same thing for perpendicular lines.
Say we need to find the equation for the line that is perpendicular to Y equals one fourth
X minus two and passes through the point negative two, nine.
Well once again, we know we need the opposite reciprocal of the slope.
That means one fourth becomes negative four.
Y equals negative four X plus B.
We plug in the coordinates of the point we have, and nine will equal negative four times
negative two plus B.
This product is eight, we subtract eight from both sides, and B equals one, so the equation
is Y equals negative four X plus one.
None of this is too tricky, but it’s important to know these rules, as these are extremely
common types of questions in algebra, so let’s check comprehension.