Slopes and equations of straight lines on the coordinate plane
The slope of a line measures its steepness. For the line through (x₁, y₁) and (x₂, y₂), the slope is m = (y₂ − y₁)/(x₂ − x₁), which also equals tan θ, where θ is the angle the line makes with the positive x-axis. Horizontal lines have slope 0 and vertical lines have an undefined slope. Two non-vertical lines are parallel when their slopes are equal and perpendicular when the product of their slopes is −1.
A line can be written in several forms. The point–slope form is y − y₁ = m(x − x₁), the slope–intercept form is y = mx + c, where c is the y-intercept, and the intercept form is x/a + y/b = 1. Every line can also be written in the general form Ax + By + C = 0, whose slope is −A/B. The perpendicular distance from a point (x₁, y₁) to this line is |Ax₁ + By₁ + C| ÷ √(A² + B²), and the distance between the parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is |C₁ − C₂| ÷ √(A² + B²).
Why it matters
Straight-line graphs describe any relationship with a constant rate of change, such as a phone bill made of a fixed rental plus a charge per minute, or the velocity of a body under constant acceleration. Slope is the first form of the idea of a derivative in calculus. This chapter supplies tools used later in conic sections, linear programming and three-dimensional geometry.
Key terms
- Slope
- The ratio of vertical change to horizontal change between two points on a line.
- Y-intercept
- The y-coordinate of the point where a line crosses the y-axis.
- General form
- The equation Ax + By + C = 0, which can represent any straight line.
- Perpendicular distance
- The shortest distance from a point to a line, measured at right angles to it.