In this lesson, the slope of a line segment connecting two point will be measured and a general formula for finding the slope will be developed and used.

The line segment shown below connects the points (1, 2) and (3, –2). As we look at this diagram from left to right, the segment slopes downward. We will define a negative number to describe the downward slope and how steeply this line segment slopes. To do this, we notice that the vertical distance (rise -- actually a "fall") between these points is -4 and the horizontal distance (run) is 2. A ratio of vertical to horizontal distances is

We say that the slope of the segment connecting the points (1, 2) and (3, –2) is – 2.

If a segment connects the point (1, 2) to the point (– 3, 0), we notice that from left to right, the segment slopes upward: the slope will be positive. We calculate the slope again, using the ratio of the vertical distance to the horizontal distance or .

To summarize, the slope is a number which describes the change between two points. It is written as a ratio of the vertical to the horizontal distances between the points and can be found by calculating these distances and making a fraction referred to as .

Generalized results about the slope between two points:

The general formula for the slope between two points is slope = .

To find the vertical and horizontal distances between the two points, sometimes this can be seen by looking at the graph, but the vertical and horizontal distances between two points can always be found by subtracting the coordinates of the two points.

We have rise = and run = giving us the vertical and horizontal distances when the coordinates of the two points are .

Important facts to remember:

1. If a line segment slopes upward from left to right, the slope is positive.
2. If a line segment slopes downward from left to right, the slope is negative.
3. If a line segment is horizontal, the slope is 0 (zero).
4. If a line segment is vertical, the slope is undefined.

Let's practice:

1. What is the slope of the segment connecting the points (3, –4) and (–2, 6)?

   We calculate the .

2. What is the slope of the segment connecting the points (3, 7) and (1, 7)?

   We calculate the . This line segment is horizontal. Any horizontal line has a slope of 0.

3. What is the slope of the segment connecting the points (4, 9) and (4, 3)?

   We calculate the  . This slope is undefined since the denominator is 0 and we cannot divide by zero. The line segment is vertical. For any vertical line, the slope is undefined.

4. If the slope of a segment connecting the points (x, 5) and (2, -4) is 6, what is the value of x?

   We calculate the .
   
   We can solve this equation by multiplying both sides by 2 - x, yielding .
   
   If , then -21 = -6x and .