Slope-Intercept Form
Slope-intercept form is the equation of a line written as y = mx + b, where m is the slope and b is the y-intercept.
Formula
y = mx + b
Definition
Slope-intercept form writes the equation of a non-vertical line as $y = mx + b$, where $m$ is the slope (steepness and direction) and $b$ is the y-intercept, the point where the line crosses the y-axis. Any linear equation can be rearranged into this form by solving for $y$, making it the most common form for graphing and for reading off a line's key features at a glance. Formally, it is the unique representation of a non-vertical affine function $f: \mathbb{R} \to \mathbb{R}$ as a degree-$1$ polynomial, where $m$ is the constant derivative $f'(x)$ and $b = f(0)$ is the initial value; the form generalizes to $y = m_1x_1 + m_2x_2 + \ldots + b$ in multiple regression, where each $m_i$ is a partial slope.
Example
$y = 2x + 3$ has slope $m = 2$ and y-intercept $b = 3$; to graph it, start at $(0, 3)$ and move up $2$, right $1$. To convert $3x - 2y = 8$ to this form, subtract $3x$ to get $-2y = -3x + 8$, then divide by $-2$: $y = \frac{3}{2}x - 4$. A cost model $C(x) = 0.15x + 25$ has marginal cost $0.15$ per item and fixed cost $25$, with the slope-intercept form isolating these economic parameters directly.
Key Insight
The form is the easiest for graphing: $b$ tells you where to start and $m$ tells you which direction to go, and it makes comparing two lines simple, matching $m$ values mean parallel lines, and $m$ values multiplying to $-1$ mean perpendicular lines. In linear algebra terms, $y = mx + b$ is an affine, not strictly linear, map because of the constant $b$; a purely linear map must satisfy $f(0) = 0$, which would require $b = 0$.