Equations of lines come in several different forms. Two of those are:

1. slope-intercept form

where m is the slope and b is the y-intercept

2. general form

Your teacher or textbook will usually specify which form you should be using. In the examples worked in this lesson, answers will be given in both forms.

When a problem asks you to write the equation of a line, you will be given certain information to help you write the equation. The strategy you use to solve the problem depends on the type of information you are given. Let’s examine these situations.

Given a Point and a Slope

When you are given a point and a slope and asked to write the equation of the line that passes through the point with the given slope, you have to use what is called the point-slope form of a line.

When using this form you will substitute numerical values for x₁, y₁ and m. You will NOT substitute values for x and y. Look at the slope-intercept and general forms of lines. Those have x and y variables in the equation.

You may be wondering why this form of a line was not mentioned at the beginning of the lesson with the other two forms. That is because the point-slope form is only used as a tool in finding an equation. It is not a way to present your answer. The slope-intercept form and the general form are how final answers are presented.

Let's Practice:

1. Find the equation of the line that goes through the point (4, 5) and has a slope of 2.

Since you have a point and a slope, you should use the point-slope form of a line. Some students find it useful to label each piece of information that is given to make substitution easier. If you are comfortable with plugging values into the equation, you may not need to include this labeling step.

Now substitute those values into the point-slope form of a line.

Now you need to simplify this expression. The process for simplifying depends on how you are going to give your answer. The process for obtaining the slope-intercept form and the general form are both shown below. Both forms involve strategies used in solving linear equations. If you need to practice these strategies, click here. (link to linear equations solving.doc)

Slope-intercept form	General Form

2. Find the equation of the line that goes through the point (-3, 2) and has a slope of -4/5.

Although the numbers are not as easy to work with as the last example, the process is still the same.

Label your information:

Plug those values into the point-slope form of the line:

Now simplify this expression into the form you need.

You can take the slope-intercept form and change it to general form in the following way.

Given Two Points

When you are given two points, it is still possible to use the point-slope form of a line. How is this possible if for the point-slope form you must have a point and a slope? Since you are given two points, you can first use the slope formula to find the slope and then use that slope with one of the given points. If you need help calculating slope, click here for lessons on slope. (linear equations slope.doc or coord geometry slope)

More Practice:

3. Find the equation of the line that passes through the points (-2, 3) and (1, -6).

The first step is to find the slope of the line that goes through those two points.

Now that you have a slope, you can use the point-slope form of a line. You also have TWO points use can use. How do you know which one is the right one? The answer is: it doesn’t matter. You can use either of the two points you have been given and you equation will still come out the same. Most students, since they have already labeled a and when finding the slope, choose to keep that labeling system. Other students will try to look ahead a few steps and see which point might be easiest to use. We will maintain the labeling we used for finding slope.

Transforming the slope-intercept form into general form gives

Parallel and Perpendicular

There is one other common type of problem that asks you to write the equation of a line given certain information. This type of problem involves writing equations of parallel or perpendicular lines. To learn more about parallel and perpendicular lines and their slopes, click here (link to coord geometry parallel)

As a quick reminder, two lines that are parallel will have the same slope. If two lines are perpendicular, their slopes are negative reciprocals of each other.

Final Practice:

4. Find the equation of the line that passes through (1, -5) and is parallel to .

As we have in each of the other examples, we can use the point-slope form of a line to find our equation. We are given the point, but we have to do a little work to find the slope.

We know we are looking for a line parallel to . That means our line will have the same slope as the line we are given. So if we can find the slope of , we will have the information we need to proceed with the problem.

If we re-write in slope-intercept form, we will easily be able to find the slope. If you need help rewriting the equation, click here for practice (link to linear equations slope.doc)

Since the slope of the given line is 2 and we want to write the equation of a line parallel, we will use slope = 2 in the point-slope form of a line.

Transforming the slope-intercept form into general form gives

5. If the problem in Example 4 had asked you to write the equation of a line perpendicular to the one given, you would begin the problem the same way. You would first find the slope of the given line, but you would then use the negative reciprocal in the point-slope form.

And in general form