Lines

A line is simply what you get when something changes at a steady rate, like a taxi fare going up the same amount every mile. The slope formula looks like a fraction of subscripts, but it's just rise over run: $y_2 - y_1$ (how far up you went between two points) divided by $x_2 - x_1$ (how far across). And $y = mx + b$ is just $m$ (the rate) times $x$, plus $b$ (where you start, the y-intercept). Lines are the simplest models there are, which is why they show up everywhere. It really is that simple.

Slope. Positive slope rises to the right, negative falls, and zero is flat. Why is the slope the same no matter which two points you pick? Because the rate is constant: walk twice as far along and you rise twice as much, so the ratio doesn't change. A vertical line has no slope at all, because the run is zero and you can't divide by zero.

Three forms. They're all the same line, just written to show different things.

• Slope-intercept $y = mx + b$: read off the slope and where it crosses the y-axis.
• Point-slope $y - y_1 = m(x - x_1)$: the fastest way to write a line through a known point. It's just the slope formula with $(x, y)$ as the second point, multiplied through.
• Standard $Ax + By = C$: solve for $y$ to find the slope, $m = -\frac AB$.

Parallel and perpendicular. Parallel lines have equal slopes, since they rise at the same rate and never meet. Perpendicular lines have slopes that are negative reciprocals, like $2$ and $-\frac12$. Picture turning a line a quarter turn: its run becomes its rise and its rise becomes a negative run, so rise over run flips over and changes sign.

Horizontal and vertical lines. $y = 3$ is horizontal (slope 0), and $x = 3$ is vertical (undefined slope).

Hourly wages, taxi fares (a flat fee plus a rate per mile), unit conversions and linear depreciation are all lines: the slope is the rate, the intercept is the starting value.