Instantly find the equation of a line in slope‑intercept form. Enter slope (m) and y‑intercept (b), or two points. Visualize the line, convert to point‑slope and standard forms, with step‑by‑step derivation.
The slope-intercept form — \( y = mx + b \) — is the most intuitive way to describe a linear relationship. The slope \( m \) represents the rate of change (rise over run), while \( b \) is the y‑intercept, where the line crosses the y‑axis. This form is fundamental in economics (cost functions), physics (velocity vs. time), and data science (linear regression).
The graph auto‑scales, marks the y‑intercept, the given points, and draws a slope triangle to visualize rise over run.
From m and b: The line is \( y = mx + b \). Point-slope form using the y‑intercept point \((0,b)\): \( y - b = m(x - 0) \). Standard form \( Ax + By = C \) is derived by rearranging and simplifying integer coefficients (with \( A > 0 \) when possible).
From two points: Compute \( m = \frac{y_2 - y_1}{x_2 - x_1} \) (handle vertical lines separately). Then calculate \( b = y_1 - m x_1 \) to get slope-intercept form. For vertical lines (\( x_1 = x_2 \)), the equation is \( x = x_1 \) (slope undefined).
A company's fixed costs are $500 (y‑intercept) and variable cost per unit is $12.50 (slope). The total cost equation is \( C = 12.5x + 500 \). Using the calculator, managers can instantly predict total cost for any production volume (x) and graph the break‑even analysis. Similarly, in physics, position \( y = vt + y_0 \) uses slope as velocity.
Engineers use slope-intercept to model voltage-current relationships (Ohm's law: \( V = IR \), where slope = resistance).
| Input Example | Slope-Intercept Form | Point-Slope Form | Standard Form |
|---|---|---|---|
| m=3, b= -2 | y = 3x - 2 | y + 2 = 3(x - 0) | 3x - y = 2 |
| Points (2,3) & (4,7) | y = 2x - 1 | y - 3 = 2(x - 2) | 2x - y = 1 |
| m=0, b=5 | y = 5 | y - 5 = 0 | y = 5 |
| Vertical: (4,1),(4,8) | Undefined slope | x = 4 | x = 4 |