Lines in the Plane and Slope

Linear equations relate two variables and are named for their graph, which is a straight line.
Slope-Intercept Form: The equation $y = mx + b$ represents a line with slope $m$ and y-intercept $(0, b)$ .
Slope ( $m$ ): Represents the vertical change (rise) per unit of horizontal change (run) from left to right.
Vertical Lines: Equations follow the form $x = a$ . The slope is undefined because division by zero is not possible.
Horizontal Lines: Equations follow the form $y = b$ . The slope is zero ( $m = 0$ ).

Ratio: If the $x$ - and $y$ -axes use the same unit of measure, the slope has no units and is a ratio.
Rate of Change: If the axes use different units, the slope represents a rate.
Marginal Cost: In economics, the slope of a linear cost equation represents the extra cost of producing one additional unit. In the model $C = 25x + 3500$ , the marginal cost is $25$ .
Fixed Cost: The y-intercept represents costs incurred regardless of production volume ( $3500$ in the example above).

The slope $m$ of a nonvertical line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
The Greek letter delta ( $\Delta$ ) denotes change.
Consistency in the order of subtraction for both numerator and denominator is required to find the correct slope.

General form: $Ax + By + C = 0$
Vertical line: $x = a$
Horizontal line: $y = b$
Slope-intercept form: $y = mx + b$
Point-slope form: $y - y_1 = m(x - x_1)$ . This form is useful for finding the equation of a line when given a point and a slope or two points.

Linear Extrapolation: Estimating a data point that lies outside the range of given points.
Linear Interpolation: Estimating a data point that lies between two given points.
Example (Starbucks Corporation): Using sales per share of $\$15.71$ (2011) and $\$17.75$ (2012) to create a linear model for future estimation.
Linear Depreciation: Also called straight-line depreciation; the book value of an asset decreases by the same amount each year.
Example (Equipment): A machine costing $\$12,000$ with a salvage value of $\$2000$ over $8$ years has an annual depreciation rate (slope) of $-\frac{1250}{1}$ . Equation: $V = -1250t + 12000$ .

Parallel Lines: Two nonvertical lines are parallel if and only if their slopes are equal ( $m_1 = m_2$ ).
Perpendicular Lines: Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals ( $m_1 = -\frac{1}{m_2}$ ).