Slope-Intercept Form
AlgebraSlope-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 is a way to write the equation of a straight line: $y = mx + b$. The letter $m$ is the slope (steepness), and $b$ is where the line crosses the y-axis.
Example
$y = 2x + 3$ has slope $m = 2$ (goes up $2$ for every $1$ right) and y-intercept $b = 3$ (crosses y-axis at $3$). To graph it, start at $(0, 3)$ and move up $2$, right $1$.
Key Insight
Slope-intercept form is the easiest form for graphing: $b$ tells you where to start, and $m$ tells you which direction to go.
Definition
Slope-intercept form $y = mx + b$ expresses a non-vertical line using its slope $m$ and y-intercept $b$ directly. Any linear equation can be rearranged into this form by solving for $y$. It is the most common form for graphing and for identifying key features of a line at a glance.
Example
Convert $3x - 2y = 8$ to slope-intercept form: subtract $3x$, getting $-2y = -3x + 8$; divide by $-2$, giving $y = \frac{3}{2}x - 4$. Slope is $3/2$, y-intercept is $-4$.
Key Insight
Slope-intercept form makes it simple to compare two lines: if $m$ values match, the lines are parallel; if $m$ values multiply to $-1$, they are perpendicular.
Definition
Slope-intercept form $y = mx + b$ is the unique representation of a non-vertical affine function $f: \mathbb{R} \to \mathbb{R}$ as a degree-$1$ polynomial. The coefficient $m$ is the derivative $f'(x)$ (constant for all $x$), 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
A cost model $C(x) = 0.15x + 25$ ($x$ items produced) has marginal cost $0.15$ per item and fixed cost $25$. The slope-intercept form isolates these economic parameters directly.
Key Insight
In linear algebra, $y = mx + b$ is an affine (not strictly linear) map because of the constant $b$. A purely linear map satisfies $f(0) = 0$, which would require $b = 0$.