How do you calculate the slope of a line?

Definition

The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change ($Delta y$) to the horizontal change ($Delta x$) between two points on the line. If the points are $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Worked Example

Find the slope of the line passing through the points $(2, 3)$ and $(5, 11)$.

Step 1: Identify the coordinates:

• $(x_1, y_1) = (2, 3)$
• $(x_2, y_2) = (5, 11)$

Step 2: Substitute into the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

$$ m = \frac{11 - 3}{5 - 2} $$

Step 3: Calculate the differences:

• $y_2 - y_1 = 11 - 3 = 8$
• $x_2 - x_1 = 5 - 2 = 3$

Step 4: Compute the slope:

$$ m = \frac{8}{3} $$

Takeaways

• The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• A positive slope means the line rises as it moves right; a negative slope means it falls.
• The slope quantifies how much $y$ changes for each unit increase in $x$.