This chapter covers determining and calculating the slope of a line.

EO 1.14 STATE the definition of the following terms: a. Slope

b. Intercept

EO 1.15 Given the equation, CALCULATE the slope of a line.

EO 1.16 Given the graph, DETERMINE the slope of a line.

Many physical relationships in science and engineering may be expressed by plotting a straight line. The slope(m), or steepness, of a straight line tells us the amount one parameter changes for a certain amount of change in another parameter.

Slope

For a straight line, slope is equal to rise over run, or

Consider the curve shown in Figure 11. Points P1 and P2 are any two different points on the line, and a right triangle is drawn whose legs are parallel to the coordinate axes. The length of the leg parallel to the x-axis is the difference between the x-coordinates of the two points and is called "x," read "delta x," or "the change in x." The leg parallel to the y-axis has length Ay, which is the difference between the y-coordinates. For example, consider the line containing points (1,3) and (3,7) in the second part of the figure. The difference between the x-coordinates is x = 3-1 = 2. The difference between the y-coordinates is y = 7-3 = 4. The ratio of the differences, y/x, is the slope, which in the preceding example is 4/2 or 2. It is important to notice that if other points had been chosen on the same line, the ratio y/x would be the same, since the triangles are clearly similar. If the points (2,5) and (4,9) had been chosen, then y/x = (9-5)/(4-2) = 2, which is the same number as before. Therefore, the ratio y/x depends on the inclination of the line, m = rise [vertical (y-axis) change] - run [horizontal (x-axis) change].

Figure 11 Slope

Since slope m is a measure of the steepness of a line, a slope has the following characteristics:

1. A horizontal line has zero slope.

2. A line that rises to the right has positive slope.

3. A line rising to the left has negative slope.

4. A vertical line has undefined slope because the calculation of the slope would involve division by zero. (y/x approaches infinity as the slope approaches vertical.)

Example: What is the slope of the line passing through the points (20, 85) and (30, 125)?

Solution:

Given the coordinates of the y-intercept where the line crosses the y-axis [written (0, y)] and the

equation of the line, determine the slope of the line.

The standard linear equation form is y = mx + b. If an equation is given in this standard form, m is the slope and b is the y coordinate for the y-intercept.

Example: Determine the slope of the line whose equation is y = 2x + 3 and whose y-intercept is (0,3).

Solution:

y = mx + b

y=2x+3

m=2

Example: Determine the slope of the line whose equation is 2x + 3y = 6 and whose y-intercept is (0,2).

Solution:

Example:

Plot the graph of the following linear function. Determine the x-intercept, the y-intercept, and the slope.

7x+3y=21

Solution:

Summary

The important information in this chapter is summarized below.

Slopes Summary

For a straight line, slope is equal to rise over run, or

Since slope m is a measure of the steepness of a line, a slope has the following characteristics:

1. A horizontal line has zero slope.

2. A line that rises to the right of vertical has positive slope.

3. A line rising to the left of vertical has negative slope.

4. A vertical line has undefined slope because the calculation of the slope would involve division by zero (y/x approaches infinity as the slope approaches vertical).