A line in two-dimensional Cartesian space has the form $$Ax + By + C = 0$$, where $$A$$ and $$B$$ are not both 0. Solving for $$y$$ yields $$y = -{A \over B}x - {C \over B},~~ B \ne 0$$ This is sometimes called the slope-intercept form of the line because $$-{A \over B}$$ represents the slope, and $$-{C \over B}$$ represents the $$y$$-intercept of the line. (Note that if $$B = 0$$, the line has no $$y$$-intercept. It is a vertical line that passes through $$x = -{C \over A}$$ on the $$x$$ axis. If $$C = 0$$ as well, then $$x = 0$$, a vertical line that passes through every point on the $$y$$ axis.) The slope-intercept form of the line is often written as $$y = mx + b$$ where $$m$$ represents the slope, and $$b$$ represents the $$y$$-intercept.

The slope of a line describes how much the line slants and in what direction. A horizontal line has no slope ($$m = 0$$). A vertical line has infinite slope ($$m = {\infty}$$). A line with positive slope ($$m \gt 0$$) ascends from left to right. A line with negative slope ($$m \lt 0$$) descends from left to right. The slope is the ratio between the change in $$y$$ and the corresponding change in $$x$$. For example, a line with a slope of 2 ($$m = 2$$) rises 2 steps for every 1 step to the right. Likewise, a line with a slope of −⅓ ($$m = -{1 \over 3}$$) falls 1 step for every 3 steps to the right.

Since the slope is the change in $$y$$ over the change is $$x$$, we can use any two points on the line to find it. Given any two points $$P = (x_1, y_1)$$ and $$Q = (x_2, y_2)$$, the slope of the line containing those points will be $$m = {{y_1 - y_2} \over {x_1 - x_2}},~~ (x_1 - x_2 \ne 0)$$ Once we know the slope, the $$y$$-intercept follows immediately. $$b = y_1 - mx_1$$

For example, Given the points $$P = (6, 0)$$ and $$Q = (0, 2)$$, then $$m = {{0 - 2} \over {6 - 0}}, m = -{1 \over 3}$$ and $$b = 0 + {1 \over 3} \times 6, b = 2$$. The equation of the line is therefore $$y = -{1 \over 3}x + 2$$ or in terms of the original form we started with, $$x + 3y - 6 = 0$$