## Working with Lines

This page contains examples of common calculations involving
the graphs and equations of lines .

### 1. Given the General Form, Find Everything

Consider the line defined by

To find the intercept we set and solve the resulting
equation:

The
-intercept is . Similarly, to find the -intercept we
set and solve the resulting equation:
The -intercept is
. The and -intercepts also gives us two
points, namely and . Two
points give us the slope:
We
already know the -intercept so the slope-intercept form of the
line is
Of course, we can also find
that form by solving the equation directly for . Subtract
on both sides and divide by negative to get the same
equation.
The line is shown in the following Figure:

### 2. Given Two Points, Find the Line

Suppose we know that our line contains the points and .
Then we can immediately compute the slope:

Thus the slope -intercept form of our line is
where we not yet know . However, we know two points on the line.
Either can be used to compute . In fact, it is a good idea to use both points to check our calculation. (Always check your answers!). Using the point
we obtain the equation
The other point, , gives the same value of :
The following Figure shows the required line:

### 3. Given Two Intercepts, Find the Line

Two intercepts are a special case of two points. For example, suppose
the intercept is and the intercept is . Then
we know that the graph of the line contains the points and
. The slope is

The "intercept" in the slope-intercept form of the line is the
intercept, thus the slope intercept form of this particular line is
This particular line is shown in the following
Figure:

### 4. Finding the Intersection of Two Lines

The key here is the fact that the coordinates of the intersection
satisfy the equations of both lines. Suppose we have the lines

Both equations hold for the intersection . Since the
values are equal we obtain the equation
Adding and on both sides gives
Dividing by gives
To obtain we substitute this value of in one of the equations, and check that we get the same answer by substituting in the other equation.
We get
So the intersection point is . The following Figure shows both lines and the intersection:

### 5. Finding a Perpendicular Line

Suppose we are given a line and a point . We want an
equation of the line through and perpendicular to . The
key fact here is that lines are perpendicular if their slopes are
negative reciprocals of each other. For example, suppose has
the equation

and we want to find the equation of the
perpendicular line that passes through the point . The
slope of that line is the negative reciprocal of , i.e.,
and so the perpendicular line has the equation
where we need to determine . Since
lies on that perpendicular line we have the equation
and so . The equation of the
perpendicular line is
The
following Figure shows both lines:

As described above, we can also compute their
intersection. Solving

gives
and substituting in one or both of the equations gives
. The two lines intersect in the point . Some of
the home work problems ask to compute the distance of from the
intersection point, in this case that distance is
### 6. Finding the Distance between a Point and a Line

The procedure is outlined in the preceding paragraph. Here is a n
outline. Suppose the line is and the point is .

- Find an equation of the line through , and perpendicular to .

- Compute the intersection of and .

- Compute the Distance between and .