The slope of the line which passes through the points and is
The slope measures the rate at which a line goes up or down as you move to the right. For example, a line with slope 4 goes up 4 units for every 1 unit you move to the right. A line with slope -2 goes down 2 units for every 1 unit you move to the right.
Example. Find the slope of the line which passes through and .
Example. A line has slope 3. If you move 2 units to the right, how far up or down does the line go?
Example. A line has slope . Does it go from northwest to southeast or from southwest to northeast?
It goes from northwest to southeast.
Example. Find the slope of the line which passes through and .
A line with slope 0 is horizontal --- parallel to the x-axis.
Example. Find the slope of the line which passes through and .
If I use the slope formula, I get
A vertical line has undefined slope.
A line may be represented by various equations. Here are a few important forms:
You can get from one form to another using algebra. For example:
The last equation is in form, with , , and .
Example. Which of the following equations represent lines?
is a line, as is .
is not a line, because of the term.
is almost a line --- in fact, if I multiply both sides by x, I get , which is a line. But the original equation is a line minus one point, because plugging in would cause division by 0.
Consider the form
This is called point-slope form. It is the equation of a line with slope m which passes through the point .
Example. Find the equation of the line with slope -17 which passes through the point .
Example. Find the slope of the line . Find at least two different points on the line.
The slope is . And from the fact that the equation is in point-slope form, I can see that the line passes through the point .
To get another point on the line, plug in a random value for x and solve for y. For example, if I use , I get
Therefore, the point also lies on the line.
Example. Find the equation of the line which passes through the points and .
First, I'll find the slope:
Using the point and point-slope form, I get the equation
If I used the point , I'd get
In fact, the two forms are equivalent:
Example. Find the equation of the horizontal line which passes through the point . Find the equation of the vertical line which passes through the point .
A horizontal line has slope 0, so the equation of the horizontal line which passes through the point is
A vertical line has undefined slope. But if you draw the picture, you can see that the vertical line passing through is :
This works in general. For example, the horizontal line passing through is . The vertical line passing through is .
The y-intercept of a line is the point where the line intersects the y-axis. Likewise, the x-intercept is the point where the line intersects the x-axis.
The form
is called slope-intercept form. In this equation, m is the slope of the line and b is the y-intercept.
Example. Find the equation of the line with slope 15 and y-intercept -32.
Example. Find the slope and y-intercept of the line .
The slope is -17, while the y-intercept is .
Example. Find the slope and y-intercept of the line .
I use algebra to put the equation into slope-intercept form:
The slope is and the y-intercept is .
Example. Find the point where the lines and intersect.
To find where two graphs intersect, solve their equations simultaneously.
From , I get . Plug this into :
Then . The point of intersection is .
Example. Find the x-intercept of the line .
To find the x-intercept, set and solve for x. The reason this works is that is the x-axis, and the x-intercept is the place where the line intersects the x-axis. By the rule of thumb from the last problem, I therefore solve and simultaneously:
The x-intercept is .
In the same way, you can find the y-intercept by setting and solving for y.
For example, 5 and are negative reciprocals of one another. So are and . Two numbers are negative reciprocals if their product is -1:
Of course, any horizontal line is perpendicular to any vertical line.
Example. Determine whether the following lines are parallel, perpendicular, or neither:
I find the slopes by putting the lines into slope-intercept form:
The slopes of the lines are and . The slopes aren't equal, so the lines aren't parallel. Since
the slopes are negative reciprocals. Hence, the lines are perpendicular.
Example. Find the equation of the line which passes through the point and is parallel to the line .
Find the slope of the given line:
The given line has slope . The line I want is parallel to the given line, so it also has slope . Since my line passes through the point , it has point-slope form
Example. Find the equation of the line which is perpendicular to the line and passes through the point .
Find the slope of the given line:
The given line has slope . The line I want is perpendicular to the given line, so my line must have slope -12 (the negative reciprocal of ). Since my line passes through , its equation is
Example. Find the equation of the line which passes through the point and is parallel to the line which passes through and .
The slope of the line which passes through and is
The line I want is parallel to this line, so it also has slope . Since my line passes through , its point-slope form is
Copyright 2008 by Bruce Ikenaga