Straight Line
Easy Overview
Remember drawing straight lines on graph paper in school? It turns out there's a whole universe hiding inside those lines. Slope, intercepts, parallel lines, perpendicular lines โ this chapter gives you the tools to describe any straight line with an equation. And once you can do that, you can predict where lines meet, how steep they are, and a whole lot more.
Slope of a line
Slope (m) tells you how steep a line is. It's rise over run โ change in y divided by change in x. Between any two points (xโ, yโ) and (xโ, yโ), m = (yโ โ yโ)/(xโ โ xโ). A positive slope tilts up, negative tilts down. Zero slope? Flat horizontal line. Undefined slope? Vertical line. That's all there is to it.
Equation of a line in all its forms
There are way too many forms, and you need 'em all. Slope-intercept: y = mx + c. Point-slope: y โ yโ = m(x โ xโ). Two-point: (y โ yโ)/(x โ xโ) = (yโ โ yโ)/(xโ โ xโ). Intercept form: x/a + y/b = 1. General form: ax + by + c = 0. Which one to use? Depends what info you're given. If you have slope and y-intercept, do slope-intercept. If you have two points, use two-point form. Easy.
Parallel and perpendicular lines
Two lines are parallel if their slopes are equal: mโ = mโ. They never meet โ like train tracks. Perpendicular lines meet at a right angle, and their slopes multiply to โ1: mโยทmโ = โ1. So if one line has slope 2, a perpendicular line has slope โ1/2. Flip it, negate it โ done.
Distance between a point and a line
Need to find how far a point is from a line? There's a formula. For a line ax + by + c = 0 and point (xโ, yโ), distance = |axโ + byโ + c| / โ(aยฒ + bยฒ). That absolute value makes sure distance is always positive. The denominator is just Pythagoras on the coefficients. Give it the coordinates and the line, and it spits out the perpendicular distance.
Family of lines
A 'family of lines' means lines that share something in common. Like all lines passing through a given point, or all lines with the same slope. The general trick: if you have two lines Lโ = 0 and Lโ = 0, then Lโ + ฮปLโ = 0 represents a family of lines through their intersection. Change ฮป and you get different lines, but they all pass through that same intersection point.
Key Points
- โขSlope m = (yโ โ yโ)/(xโ โ xโ) = rise/run
- โขy = mx + c โ slope-intercept form
- โขParallel lines: mโ = mโ
- โขPerpendicular lines: mโยทmโ = โ1
- โขPoint-to-line distance: |axโ + byโ + c| / โ(aยฒ + bยฒ)
- โขx-intercept is where y = 0, y-intercept is where x = 0
Practice Questions
- Find the equation of a line passing through (2, 3) with slope 4.
- Find the distance between the parallel lines 3x + 4y โ 7 = 0 and 3x + 4y + 8 = 0.
- Find the equation of a line perpendicular to 2x โ 3y + 5 = 0 and passing through (1, โ1).
- Show that three given points are collinear using slope.
- A line makes intercepts whose sum is 7 and product is 12. Find its equation.