# Slope Of A Line With Two Points

This topic is about the slope of a line. This topic is typically found in (Canadian) high school mathematics.

Suppose we are given a point with the co-ordinate (, ) and a point with the co-ordinate (, ). The line segment connects the point to the point .

The slope of a line represents the ratio of the rise over the run. The rise represents the vertical distance between two points and the run represents the horizontal distance (left to right) between two points.

To compute the slope for two points (in the xy-plane), we use the formula:

where is greater than as we go from left to right in the co-ordinate system.

Here is a visual as a guide:

Source: http://www.mathwarehouse.com/algebra/linear_equation/images/slope-of-a-line2.gif

Slope Cases

The slope value can take on many values but we look at some general cases.

If the value is positive or , we have an upward slope. A value which is positive but closer to zero would be a flat upward sloping line. If the value is a much larger positive number then the upward sloping line would be steeper and would look closer to a vertical line.

If the value is negative or , we have an downward slope. A value which is negative but closer to zero would be a flat downward sloping line. If the value is a much larger negative number then the negative sloping line would be steeper and would look closer to a vertical line.

When the value is zero, the slope is neither positive nor negative. There is no rise in the run. The line from point to point is a flat horizontal line.

If then in the denominator of the slope. A fraction with division by zero is undefined and thus the slope is undefined.

The following picture provides a summary of the cases.

Source: https://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U04_L1_T1_text_final_files/image014.gif

Examples

Example One

Suppose that the point is (0, 3) and the point is (1, 2). The slope from point to point is as follows:

The slope of the line from to is downward sloping. For every increase in by 1, decreases by 1.

Example Two

Given the point being (1, 4) and the point as (5, 7), the slope from point to point is:

The slope of the line from to is upward sloping. For every increase in by 1, decreases by .