Slope Intercept Form Calculator

The slope-intercept form of a linear equation in two variables is:

$y = mx + b$

where:

• $m$ is the slope — how steeply the line rises or falls. Slope $= \dfrac{\text{rise}}{\text{run}}$.
• $b$ is the y-intercept — the $y$-value where the line crosses the y-axis (the point $(0, b)$).

Why this form is special: it reads off two pieces of geometric information at a glance — the slope and the y-intercept — without any computation. By contrast, standard form $Ax + By = C$ obscures both.

Slope-intercept is the working form of choice for graphing lines, comparing parallel/perpendicular relationships, and writing equations from a description.

Case 1: From an Equation in Standard Form

Given $Ax + By = C$, solve for $y$:

$By = -Ax + C \quad \Rightarrow \quad y = -\frac{A}{B}x + \frac{C}{B}$

So $m = -A/B$ and $b = C/B$.

Case 2: From Two Points

Given $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Then use one of the points to solve for $b$:

$b = y_1 - m x_1$

Case 3: From Slope and One Point

Given slope $m$ and a point $(x_0, y_0)$:

$b = y_0 - m x_0$

Case 4: From a Graph

Read the y-intercept directly from where the line crosses the y-axis. Pick another lattice point and count $\text{rise} / \text{run}$ to find $m$.

Special Cases

• Horizontal line $y = c$: slope $m = 0$, y-intercept $b = c$.
• Vertical line $x = c$: slope is undefined. Cannot be written as $y = mx + b$.

Parallel and Perpendicular Lines

Two lines $y = m_1 x + b_1$ and $y = m_2 x + b_2$ are:

• Parallel iff $m_1 = m_2$ (same slope, different intercepts)
• Perpendicular iff $m_1 m_2 = -1$ (negative reciprocal slopes)

• Slope sign errors: $m = (y_2 - y_1)/(x_2 - x_1)$. Subtract $y$'s in the same order as $x$'s. Reversing one but not the other flips the sign.
• Dividing by zero: If $x_1 = x_2$, the line is vertical — slope undefined, no slope-intercept form exists.
• Confusing y-intercept with x-intercept: $b$ is the y-intercept. The x-intercept is found by setting $y = 0$ and solving for $x$.
• Forgetting to divide by $B$: When converting $Ax + By = C$ to slope-intercept, you must divide every term by $B$, not just the $y$ term.
• Wrong perpendicular slope: Perpendicular means $m_1 m_2 = -1$, so $m_2 = -1/m_1$. Just flipping the sign or just reciprocating is not enough.