Slope-intercept form calculator

What is a slope-intercept form calculator?

A slope-intercept form calculator builds the equation of a straight line from two points that the line passes through. It returns the line written in slope-intercept form $y = mx + b$ — the most common way to describe a line in algebra — together with the slope $m$ and the y-intercept $b$ on their own.

Two distinct points are enough to fix a line completely, so from them this calculator can recover both numbers that define the slope-intercept equation: how steep the line is and where it crosses the vertical axis.

Key concepts

• Point $(x, y)$ — an ordered pair that locates a position on the coordinate plane.
• Slope (m) — how steep the line is, the vertical change divided by the horizontal change between the two points.
• y-intercept (b) — the value of $y$ where the line crosses the vertical axis, i.e. where $x = 0$.
• Slope-intercept form — $y = mx + b$, the line written so that its slope and intercept can be read off directly.

How does the calculator work?

First find the slope from the two points $(x_1, y_1)$ and $(x_2, y_2)$ as rise over run:

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

Then use either point with the slope to recover the y-intercept by solving $y = mx + b$ for $b$:

$$b = y_1 - m x_1$$

Finally, the line is written in slope-intercept form:

$$y = mx + b$$

Enter the coordinates of the two points and the calculator immediately returns $m$, $b$, and the full equation. If $x_1 = x_2$ the two points lie on a vertical line, which has no defined slope and cannot be written as $y = mx + b$ — the calculator leaves the results blank in that case.

Worked examples

Example 1: line through the origin

For the points $(0, 0)$ and $(2, 4)$:

$$m = \frac{4 - 0}{2 - 0} = 2, \quad b = 0 - 2 \cdot 0 = 0$$

The equation is $y = 2x$. A line through the origin has a y-intercept of $0$.

Example 2: positive intercept

For the points $(0, 3)$ and $(2, 7)$:

$$m = \frac{7 - 3}{2 - 0} = 2, \quad b = 3 - 2 \cdot 0 = 3$$

The equation is $y = 2x + 3$. The line crosses the vertical axis at $y = 3$.

Example 3: negative slope

For the points $(1, 5)$ and $(3, 1)$:

$$m = \frac{1 - 5}{3 - 1} = -2, \quad b = 5 - (-2) \cdot 1 = 7$$

The equation is $y = -2x + 7$. The line falls two units for every unit it moves to the right.

Example 4: horizontal line

For the points $(1, 2)$ and $(4, 2)$:

$$m = \frac{2 - 2}{4 - 1} = 0, \quad b = 2 - 0 \cdot 1 = 2$$

The equation is $y = 0x + 2$, i.e. $y = 2$. Both points share the same $y$, so the line is horizontal.

Practical uses

• Algebra and graphing — read off the slope and intercept directly to sketch the line by hand.
• Statistics — express a fitted regression line as $y = mx + b$, where the slope is the average change in $y$ per unit change in $x$.
• Physics — turn two measured data points into a linear model, for example position against time at constant velocity.
• Geometry problems — once you have the slope from the slope calculator or a point from the midpoint calculator, this calculator gives the line’s full equation; for a single point and a known slope use the point-slope form calculator instead.

Notes

• The order of the two points does not matter: swapping $(x_1, y_1)$ and $(x_2, y_2)$ negates both the rise and the run, leaving the slope unchanged.
• A vertical line has no slope-intercept form. Its equation is simply $x = x_1$, and the calculator leaves the results empty.
• A horizontal line has slope $0$, so $b = y_1$ and the equation reduces to $y = b$.
• The two points must be different. If both points are identical, infinitely many lines pass through them and the line is not determined.