Equation of a Line in a Plane
Big Idea: Every straight line in a coordinate plane can be described by an algebraic equation. This equation is a rule that connects the $x$- and $y$-coordinates of every point on that line.
1. The Slope–Intercept Form
The most common form of a line's equation is:
$$y = mx + b$$
- $m$ — the slope (gradient) of the line. It tells you how steep the line is and in which direction it tilts.
- $b$ — the $y$-intercept, the point where the line crosses the $y$-axis (i.e., the value of $y$ when $x = 0$).
Understanding Slope
The slope $m$ is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$$
| Slope Value | Line Behavior |
|---|---|
| $m > 0$ | Line rises from left to right ↗ |
| $m < 0$ | Line falls from left to right ↘ |
| $m = 0$ | Horizontal line → |
| $m$ undefined | Vertical line ↑ |
Key Insight: A vertical line cannot be written in slope–intercept form. Its equation is simply $x = a$, where $a$ is the $x$-coordinate of every point on the line.