Slope

The slope of a line measures the dependent variable’s change that’s associated with a change in the independent variable. The dependent variable is the y-variable, and the independent variable is x-variable. So, the equation for the slope, which is represented by an m, of a line is:

Slope Equation - 1.png

In the above equation, the Δ is a Greek symbol which means the change in. You would read the above equation as the change in y divided by the  Δx. To find the change in x  and the Δselect two points on the graph and calculate the difference between them. That makes the equation for the slope of a line is:

Slope Equation - 8.png

Take the following figure as an example:

Slope Equation - 2.png

 

 

 

 

The two coordinates for our graph are (3, 4) and (-3, 0). It doesn’t matter which of the two points we choose Point 1 and Point 2. But once we have made that decision, we have to keep with that decision until we find the slope of the line.

For example, we are going to chose (3, 4) as Point 1 and (-3, 0) as Point 2. That makes y2 = 0, y1 = 4, x2 = -3 and x1 = 3.  Plugging those values into the slope equation yields:

Slope Equation - 9

Subtracting the two values in the numerator and the two values in the denominator.

Slope Equation - 10

Since both of the terms are negative, they cancel each other out.

Slope Equation - 11

Both of the terms are multiples of 2 so, dividing the numerator and denominator gives us the value of our slope.

Slope Equation - 12

If we chose the opposite points for Point 1 and Point 2 That would make  y2 = 4, y1 = 0, x2 = 3 and x1 = -3.  Plugging those values into the slope equation yields:

Slope Equation - 3

minus 0 is 4, and minus -3 is 6. 

Slope Equation - 4

Both of these terms are multiples of 2, so we can divide both the numerator and the denominator by gives us the slope of the line.

Slope Equation - 5

Both ways to calculate the slope of a line yields the same result.