Line Equation Solver

Math Geometry • Coordinate Geometry

Written by STEM Calculators Team Published January 23, 2026 Updated February 24, 2026

Find the equation of a line from two points, slope + point, slope + intercept, or x/y-intercepts. The graph is interactive: drag to pan, wheel/trackpad to zoom, pinch on touch.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Mode:

Inputs

\(x_1\):

\(y_1\):

\(x_2\):

\(y_2\):

Slope \(m\):

Intercept \(b\):

x-intercept \(a\):

y-intercept \(b\):

Tip: In Two points mode, slope is \(m=\dfrac{y_2-y_1}{x_2-x_1}\). If \(x_1=x_2\), the line is vertical: \(x=x_1\).

Graph options

Axis units label: Show grid: Show tick labels:

The tick spacing is adaptive to the zoom level to keep labels clean (no crowding).

Ready

Choose a mode and click Calculate.

Line diagram (pan/zoom enabled)

Drag to pan • wheel/trackpad to zoom • pinch on touch • “Reset view” fits the geometry.

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Theory

What this calculator does

A straight line in the plane can be written in several equivalent forms. This tool converts between the most common ones and handles special cases like vertical and horizontal lines.

Core formulas

1) Slope–intercept form

\[ y = mx + b \] where \(m\) is the slope and \(b\) is the \(y\)-intercept.

2) Point–slope form

\[ y - y_0 = m(x - x_0) \] This is useful when you know one point and the slope.

3) Two-point form (via slope)

Given \(A(x_1,y_1)\) and \(B(x_2,y_2)\) with \(x_1\neq x_2\), \[ m=\frac{y_2-y_1}{x_2-x_1}. \] Then you can plug \(m\) into point–slope form.

4) Standard form

\[ Ax + By = C. \] A robust way to get \(A,B,C\) from two points is: \[ A = y_2-y_1,\quad B = x_1-x_2,\quad C = Ax_1 + By_1. \] This also works for vertical lines.

5) Intercept form

If the line crosses the axes at \((a,0)\) and \((0,b)\) with \(a\neq 0\) and \(b\neq 0\), \[ \frac{x}{a}+\frac{y}{b}=1. \]

Special cases

Vertical line

A vertical line has constant \(x\) and undefined slope: \[ x = k. \] In standard form this is \(1\cdot x + 0\cdot y = k\).

Horizontal line

A horizontal line has slope \(m=0\): \[ y = b. \]

Worked example

Points \(A(0,3)\) and \(B(2,7)\).

\[ m=\frac{7-3}{2-0}=2,\qquad b = y_1 - mx_1 = 3 - 2\cdot 0 = 3. \]

So: \[ y = 2x + 3. \]

Standard form: \[ 2x - y = -3 \] (equivalent to \(y=2x+3\)).

Tips and common mistakes

If \(x_1=x_2\), do not compute slope (division by zero). The line is vertical.
When using intercepts, \(a=0\) means the line is \(x=0\) and \(b=0\) means the line is \(y=0\).
Different algebraic forms can look different but still describe the same line.

Frequently Asked Questions

How do you find the equation of a line from two points?

Compute the slope m = (y2 - y1)/(x2 - x1) if x1 is not equal to x2, then use y - y1 = m(x - x1) or y = m x + b with b = y1 - m x1. If x1 equals x2, the line is vertical and has equation x = x1.

What is the difference between slope-intercept form and point-slope form?

Slope-intercept form is y = m x + b and is convenient for graphing because b is the y-intercept. Point-slope form is y - y0 = m(x - x0) and is convenient when a point on the line and the slope are known.

What happens if the line is vertical?

A vertical line has undefined slope because x2 - x1 = 0, so the slope formula would divide by zero. The equation is written as x = k, where k is the constant x-value of every point on the line.

How do you write a line equation from x-intercept and y-intercept?

If the line crosses the axes at (a, 0) and (0, b) with a and b not equal to 0, the intercept form is x/a + y/b = 1. This can be rearranged into other equivalent forms if needed.

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Frequently Asked Questions

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