# Mathematics 1010 online

## 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 .

1. Find an equation of the line through , and perpendicular to .
2. Compute the intersection of and .
3. Compute the Distance between and .