Equation of a Straight Line

The slope (also called gradient) of a straight line measures how steeply it rises or falls. If a line passes through two points (x1, y1) and (x2, y2), its slope is m = (y2 - y1) / (x2 - x1), often read as 'rise over run'. The slope also equals tan theta, where theta is the angle the line makes with the positive direction of the x-axis (its inclination). A line going up to the right has a positive slope; a line going down to the right has a negative slope. A horizontal line has slope 0, and a vertical line has an undefined slope because the run is zero.

Every non-vertical straight line can be written as y = mx + c. Here m is the slope and c is the y-intercept, the value of y where the line crosses the y-axis (the point where x = 0). This is the most useful form: just by looking at the equation you can read off both the steepness (m) and the starting height (c). For example, y = 2x + 3 is a line with slope 2 that crosses the y-axis at (0, 3). To find the slope of any line given in the form ax + by + c = 0, rearrange it into y = mx + c and read off m.

Often you are not given the equation but some geometric information. If you know one point (x1, y1) on the line and its slope m, use the point-slope form y - y1 = m(x - x1). If you know two points on the line, first find the slope from the two points, then use the point-slope form with either point. The two-point form (y - y1)/(x - x1) = (y2 - y1)/(x2 - x1) does both steps at once. After finding the equation you can rearrange it into y = mx + c or the general form ax + by + c = 0.

Slope decides the relationship between two lines. Two lines are parallel exactly when they have the same slope: m1 = m2. Two lines are perpendicular exactly when the product of their slopes is -1: m1 × m2 = -1, which means each slope is the negative reciprocal of the other. So a line perpendicular to one with slope 2 has slope -1/2. These two rules let you write the equation of a line parallel or perpendicular to a given line through a given point: keep the same slope for parallel, take the negative reciprocal for perpendicular, then apply the point-slope form.