Point-slope form calculator

What is a point-slope form calculator?

A point-slope form calculator builds the equation of a straight line when you know just two things: a single point that the line passes through and the line’s slope. From those inputs it produces the line written in point-slope form, the same line rewritten in the more familiar slope-intercept form $y = mx + b$, and the y-intercept $b$ itself.

This is one of the quickest ways to nail down a line in coordinate geometry. You do not need two points or a graph — one point and the steepness are enough to fix the line completely.

Key concepts

• Point $(x_1, y_1)$ — a known location the line passes through.
• Slope (m) — how steep the line is, the vertical change per unit of horizontal change.
• Point-slope form — $y - y_1 = m(x - x_1)$, the direct expression of “a line through $(x_1, y_1)$ with slope $m$”.
• y-intercept (b) — the value of $y$ where the line crosses the vertical axis, i.e. where $x = 0$.

How does the calculator work?

Start from the point-slope form, which is true for any point on the line:

$$y - y_1 = m(x - x_1)$$

To get the slope-intercept form, solve for $y$:

$$y = mx - m x_1 + y_1$$

The constant term is the y-intercept, so:

$$b = y_1 - m x_1$$

Enter $x_1$, $y_1$, and the slope $m$, and the calculator immediately returns $b$ along with both equation forms. If any of the three inputs is missing the result is left blank, because a single line cannot be determined yet.

Worked examples

Example 1: positive slope

For the point $(2, 3)$ with slope $m = 4$:

$$b = y_1 - m x_1 = 3 - 4 \cdot 2 = -5$$

The line is $y = 4x - 5$, and in point-slope form $y - 3 = 4(x - 2)$.

Example 2: negative slope

For the point $(1, 5)$ with slope $m = -2$:

$$b = 5 - (-2) \cdot 1 = 5 + 2 = 7$$

The line is $y = -2x + 7$, and in point-slope form $y - 5 = -2(x - 1)$.

Example 3: line through the origin

For the point $(0, 0)$ with slope $m = 3$:

$$b = 0 - 3 \cdot 0 = 0$$

The line is $y = 3x$. Any line through the origin has a y-intercept of $0$, so the point-slope and slope-intercept forms collapse to the same simple equation.

Practical uses

• Algebra and graphing — quickly convert between the point-slope and slope-intercept descriptions of a line.
• Physics — write the equation of a motion or response that you measured at one instant, given its rate of change.
• Data and modelling — extend a known data point along a trend whose rate you already estimated.
• Geometry problems — when you have located a point with the midpoint calculator and computed a direction with the slope calculator, this calculator finishes the job by giving the line’s full equation.

Notes

• The slope must be a real number. A vertical line has no defined slope and cannot be written in point-slope or slope-intercept form; its equation is simply $x = x_1$.
• A horizontal line has slope $0$, so $b = y_1$ and the equation reduces to $y = y_1$.
• The point you supply does not have to be the y-intercept — any point on the line works, and the calculator finds $b$ for you.