What is the slope (m) of a line?

The slope of a line is the ratio of the change in y-coordinates to the change in x-coordinates of two points on the line. The change in y-coordinates is called the RISE and the the change in x-coordinates is called the RUN {to find the change in two numbers, subtract}. Sometimes we refer to the slope as the Rise/Run.

What does the slope tell us?

The slope of a linear equation with two variable terms present tells us the direction of the graph (rising when positive, falling when negative).

How can we use the slope?

When we know the slope of a line and a point on that line, we can easily graph the line by plotting the point and following the slope to another point on that line. So, the slope also gives us a set of directions from a point on a line to another point on that same line.

How can we calculate the slope is we know two points on the line?

Since the slope is a ratio, it looks like a fraction. We are computing the difference in y-coordinates for the Rise and the difference in x-coordinates for the Run. So what will this look like?

m = y₂ -y₁ / x₂ - x₁,

where the coordinates of one point are (x₁, y₁) and the coordinates of the second point are (x₂, y₂).
Remember: since we are finding differences, you MUST keep the minus signs!

Okay, let’s do an example or three:

Ex. 1 Find the slope between (2, 4) and (7, 8).

m = y₂ -y₁ / x₂ - x₁,

Choose one of the points to use first. Let’s use (7, 8) this time.
{Remember: x₂ and y₂ are coordinates FROM THE SAME POINT!}

Begin by writing the pattern with the blanks for the coordinates (MUST keep minus signs!).

- / -   Now, fill in the coordinates from (7, 8), being careful to put the y-coordinate on top and the x-coordinate on bottom:

8-/7-  Next, we fill in the coordinates from (2, 4).

8-4 / 7-2  Calculate and simplify.

m = 4/5  Since the slope is positive, the line is Rising.

Ex. 2 Find the slope between (2, –4) and (–7, 8).

m = y₂ -y₁ / x₂ - x₁,

Choose one of the points to use first. Let’s use (–7, 8) this time.
{Remember: x₂ and y₂ are coordinates FROM THE SAME POINT!}

Begin by writing the pattern with the blanks for the coordinates (MUST keep minus signs!).

- / -   Now, fill in the coordinates from (7, 8), being careful to put the y-coordinate on top and the x-coordinate on bottom:

8-/-7-  Next, we fill in the coordinates from (2, 4-).

8-(-4) / -7-2  Calculate and simplify.

m = 12/-9 = 4/3  Since the slope is positive, the line is Falling.