# 2.2. Slope of a Line

# Slope of a Line

In this section, you will learn to:

- Find the slope of a line if two points are given.
- Graph the line if a point and the slope are given.
- Find the slope of the line that is written in the form
*y*= m*x*+*b*. - Find the slope of the line that is written in the form A
*x*+ B*y*=*c*.

In the last section, we learned to graph a line by choosing two points on the line. A graph of a line can also be determined if one point and the “steepness” of the line is known. The precise number that refers to the steepness or inclination of a line is called the **slope** of the line.

From previous math courses, many of you remember slope as the “rise over run,” or “the vertical change over the horizontal change” and have often seen it expressed as:

We give a precise definition.

**Definition: **

If ( *x*_{1}, *y*_{1}) and ( *x*_{2}, *y*_{2}) are two different points on a line, then the slope of the line is

Example 2.2.1

**Solution**

*x*

_{1},

*y*

_{1}) = (-2, 3) and (

*x*

_{2},

*y*

_{2}) = (4, −1) then the slope

To give the reader a better understanding, both the vertical change, -4, and the horizontal change, 6, are shown in the above figure.

When two points are given, it does not matter which point is denoted as (*x*_{1}, *y*_{1}) and which (*x*_{2}, *y*_{2}). The value for the slope will be the same. For example, if we choose (*x*_{1}, *y*_{2}) = (4, -1) and (*x*_{2}, *y*_{2}) = (-2, 3), we will get the same value for the slope as we obtained earlier. The steps involved are as follows.

The student should further observe that if a line rises when going from left to right, then it has a positive slope; and if it falls going from left to right, it has a negative slope.

Example 2.2.2

**Solution**

*x*1,

*y*1) = (2, 3) and (

*x*2,

*y*2) = (2, -1) then the slope

**Note: **The slope of a vertical line is undefined.

Example 2.2.3

**Solution**

Alternatively, since represents the same number, the line can be drawn by starting at the point (1, 2) and choosing a rise of 3 units followed by a run of -4 units. So from the point (1, 2), we go up 3 units, and to the left 4, thus reaching the point (-3, 5) which is also on the same line. See the figure below.

Example 2.2.4

*x*+ 3

*y*= 6.

**Solution**

*x*and

*y*intercepts seems to be a good choice. The x-intercept is (3, 0), and the y-intercept is (0, 2). Therefore, the slope is

*x*and

*y.*

Example 2.2.5

*y*= 3

*x*+ 2.

**Solution**

Look at the slopes and the y-intercepts of the following lines.

The line |
Slope |
y-intercept |

y = 3x + 2 |
3 | 2 |

y = -2x + 5 |
-2 | 5 |

y = 3/2x – 4 |
3/2 | -4 |

It is no coincidence that when an equation of the line is solved for *y*, the coefficient of the *x* term represents the slope, and the constant term represents the y-intercept.

In other words, for the line *y* = *mx* + *b*, *m* is the slope, and *b* is the y-intercept.

Example 2.2.6

Determine the slope and y-intercept of the line 2*x* + 3*y* = 6.

**Solution**

*y*.

# Practice questions

**1.** Find the slope of the line passing through the following pair of points:

**a.** (2, 3) and (5, 9)

**b.** (6, -5) and (4, -1)

**c.** (-3, -5) and (-1, -7)

**2.** Determine the slope of the line from the given equation of the line:

**a.**

**b.**

**c.**

**3.** Graph the line that passes through the given point and has the given slope.

**a. **(1, 2) and *m* = -3/4

**b. **(0, 2) and *m* = -2