Lesson 3: Finding Slopes

Property

The slope of a line is the ratio

as we move from one point to another on the line.

Examples

• A trail rises 50 feet over a horizontal distance of 200 feet. The slope is $\frac{\text{change in } y}{\text{change in } x} = \frac{50}{200} = \frac{1}{4}$.
• As we move from point $(1, 2)$ to point $(5, 10)$ on a line, the y-coordinate changes by $8$ and the x-coordinate changes by $4$. The slope is $\frac{8}{4} = 2$.
• A wheelchair ramp descends 1 foot for every 12 feet of horizontal distance. The change in y is $-1$, so the slope is $\frac{-1}{12} = -\frac{1}{12}$.

Explanation

Slope tells you the steepness of a line. For every step you take to the right (change in x), how many steps do you go up or down (change in y)? A bigger number means a steeper line, while a negative number means it goes downhill.

Property

The slope of a line is given by

The symbol $\Delta$ (delta) is used in mathematics to denote change in.

Examples

• If the change in y, $\Delta y$, is 6 and the change in x, $\Delta x$, is 2, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3$.
• A line moves 4 units down ($Δy = -4$) for every 10 units it moves to the right ($Δx = 10$). The slope is $m = \frac{-4}{10} = -\frac{2}{5}$.
• For a horizontal line, the y-coordinate never changes, so $\Delta y = 0$. This means the slope $m = \frac{0}{\Delta x} = 0$, no matter the change in x.

Explanation

This is the official shorthand for slope. The letter $m$ stands for slope, and the Greek letter delta ($\Delta$) is a compact way to write 'change in'. So, $m = \frac{\Delta y}{\Delta x}$ is just a neat way of writing slope equals change in y over change in x.

Property

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope $m$ of the line is calculated using the formula:

Examples