The title of this section is a statement that you may have heard before. It's a commonsense geometric idea taken from everyday experience: You can draw exactly one straight path between two dots on a piece of paper.

But is it still true if we're talking about lines in algebra? In this chapter we have seen that lines can be described by equations. If we know only the coordinates of two points, can we always find an equation for the line that goes through them?

What do you think? How would you start to handle this question? What Thinking Tip might be useful here?

An equation for a line tells us how the x- and y- coordinates of the points on that line are related. For instance, saying that a line has equation y = 2x + 1 means that, for any point on this line, if you double its x-coordinate and add 1, you will get its y- coordinate.

In the preceding section, you saw how to make an equation for a line from its slope and its y-intercept. For instance, if we know that a line has slope 5 and y- intercept 3, we can write down an equation for that line: y = 5x + 3. In general, such an equation for a line has the form

y = [slope]x + [y-intercept]

So the question of finding an equation for a line comes apart into two pieces.

A. Can we find the slope?

B. Can we find the y-intercept?

Let's try an example. How about finding an equation for the line through (1, 8) and (3, 14)?

A. Can we find the slope? Of course! The slope is the change in y-values divided by the change in x-values between any two points on the line, so we can use these two points to find the slope. We can use the points in either order, so let's....

After studying this section, you will be able to:

Use the coordinates of two points to write an equation for the line through those points;

Use graphs and equations of lines to analyze rate of change in various real world situations.