Definition
The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change ("rise") to the horizontal change ("run") between two points on the line. The slope is commonly denoted by $m$.
Mathematically, if you have two points on a line, $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is calculated as:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
- If $m > 0$, the line rises from left to right.
- If $m < 0$, the line falls from left to right.
- If $m = 0$, the line is horizontal.
- If the denominator is $0$, the line is vertical (undefined slope). --- ## 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: %%MATH_DISPLAY_1%% %%MATH_DISPLAY_2%% %%MATH_DISPLAY_3%% **So, the slope is $frac{8}{3}$.** --- ## Takeaways - The slope of a line is calculated as $m = frac{y_2 - y_1}{x_2 - x_1}$.
- Slope describes how steep a line is and its direction.
- A positive slope means the line rises, a negative slope means it falls.