Slope
AlgebraSlope measures the steepness and direction of a line as the ratio of vertical change to horizontal change between any two points.
Formula
m = \frac{y_2 - y_1}{x_2 - x_1}
Definition
Slope tells you how steep a line is and which direction it goes. A large slope means a very steep line. A positive slope goes uphill from left to right; a negative slope goes downhill.
Example
A ski run that goes up $3$ feet for every $1$ foot forward has a slope of $3$. A ramp that drops $1$ foot over $4$ feet of length has a slope of $-1/4$.
Key Insight
Think of slope as "rise over run": how much you go up (or down) for every step you take to the right.
Definition
The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = (y_2 - y_1)/(x_2 - x_1)$, provided $x_2$ is not equal to $x_1$. Slope is constant for any two points on the same line. It represents the rate of change of $y$ with respect to $x$.
Example
Points $(1, 2)$ and $(4, 8)$: $m = (8 - 2)/(4 - 1) = 6/3 = 2$. For every $1$ unit right, $y$ increases by $2$ units.
Key Insight
Slope is a rate of change. If $y$ represents distance and $x$ represents time, slope is speed. The same concept powers calculus derivatives.
Definition
Slope is the first derivative of a linear function $f(x) = mx + b$, and equals the tangent of the angle the line makes with the positive x-axis: $m = \tan\theta$. For a line in $\mathbb{R}^2$, slope is the coefficient of $x$ in the reduced row echelon form $y = mx + b$. Parallel lines share the same slope; perpendicular lines have slopes whose product is $-1$ (when both are defined).
Example
A line with slope $m = 3/4$ makes an angle $\theta = \arctan(3/4) \approx 36.87^\circ$ with the x-axis. Its perpendicular has slope $-4/3$.
Key Insight
Slope generalizes to the gradient vector in multivariable calculus. In a linear map $\mathbb{R} \to \mathbb{R}$, slope is exactly the matrix entry (a $1 \times 1$ matrix). This connection links algebra directly to linear transformations.