Line Equation from Two Points

Enter two points to get the slope, y-intercept, and the line equation in slope-intercept, point-slope, and standard form.

Inputs

P_1(x_1,\, y_1)\quad P_2(x_2,\, y_2)

Results

ΔxΔyxyP₁P₂
\begin{aligned} m &= \dfrac{y_2 - y_1}{x_2 - x_1} \\ &= \dfrac{(11) - (2)}{(4) - (1)} \\ &= ? \end{aligned}
\begin{aligned} b &= y_1 - m x_1 \\ &= (2) - (?)(1) \\ &= ? \end{aligned}

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The equation of a line

A straight line in the coordinate plane is completely determined by any two of its points. Given points P₁ = (x₁, y₁) and P₂ = (x₂, y₂), the line through them has a single slope and can be written in three standard forms — each useful in different contexts.

The three standard forms

Slope-intercept form: y = mx + b

The most commonly used form. Here m is the slope (rise ÷ run) and b is the y-intercept (where the line crosses the y-axis).

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

b=y1−m⋅x1b = y_1 - m \cdot x_1

Best for: graphing (start at (0, b), then rise m for every 1 step right), reading off the slope at a glance, and functions-of-x problems.

Point-slope form: y − y₁ = m(x − x₁)

Uses the slope and one of the given points directly, without first computing the y-intercept.

Best for: writing the equation when you know a point and the slope, or when deriving other forms — it's the most direct statement of what a slope means.

Standard form: Ax + By = C

All three variables on one side, no fractions (when the input coordinates are integers). Convention: A ≥ 0.

Best for: linear systems (both equations in Ax + By = C form are easy to add or subtract), finding intercepts (set x = 0 → y = C/B; set y = 0 → x = C/A).

Special cases

Vertical lines

When x₁ = x₂, the two points share the same x-coordinate and lie on a vertical line. The slope is undefined (division by zero), so slope-intercept and point-slope forms don't apply. The equation is simply x = x₁.

Horizontal lines

When y₁ = y₂, the line is horizontal. The slope is 0, the y-intercept is y₁, and all three forms simplify to y = y₁.

Worked example

Given P₁ = (1, 2) and P₂ = (4, 11):

m=11−24−1=93=3m = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3

b=2−3⋅1=−1b = 2 - 3 \cdot 1 = -1

  • Slope-intercept: y = 3x − 1
  • Point-slope: y − 2 = 3(x − 1)
  • Standard: 3x − y = 1

Converting between forms

FromToMethod
Point-slopeSlope-interceptDistribute m, isolate y
Slope-interceptStandardMove mx to the left, multiply by −1 if A < 0
StandardSlope-interceptSolve for y: y = (C − Ax) / B

Integer coefficients in standard form

When the slope m = p/q is already a fraction (with p and q coprime integers), multiply every term of y = mx + b through by q:

qy=px+qb  ⟹  px−qy=−qbqy = px + qb \implies px - qy = -qb

Now A = p, B = −q, C = −qb are all integers. This calculator applies this step automatically for integer input coordinates.

Frequently Asked Questions (FAQ)

Which equation form should I use?

Slope-intercept form (y = mx + b) is best for reading off the slope and y-intercept directly, and for graphing — start at (0, b) and rise m units for every 1 unit to the right.

Point-slope form (y − y₁ = m(x − x₁)) is best when the slope and one point are known, since it writes the equation without first solving for b.

Standard form (Ax + By = C) is best for algebra and linear-systems work, and for finding both intercepts at once by setting the other variable to zero.

How do I convert between the three forms?

To go from point-slope to slope-intercept, distribute m and solve for y: y − y₁ = m(x − x₁) → y = mx − mx₁ + y₁ → y = mx + b where b = y₁ − mx₁. To go from slope-intercept to standard form, move the mx term to the left: y = mx + b → −mx + y = b, then optionally multiply through by −1 so A is positive: mx − y = −b. Any of the three forms can also be rearranged to the others with basic algebra.

What happens when the two x-coordinates are the same?

When x₁ = x₂, the two points lie on a vertical line. A vertical line has undefined slope because the run (x₂ − x₁) is zero, making the division (y₂ − y₁) ÷ 0 undefined. The equation is simply x = x₁ — the line passes through every point with that x-value. There is no y-intercept unless x₁ = 0, in which case the line is the y-axis itself.

How do I get integer coefficients in standard form?

When the slope m = p/q is a fraction (p and q integers, gcd(p, q) = 1), multiply every term of y = mx + b through by q: qy = px + qb, which rearranges to px − qy = −qb. All three coefficients A = p, B = −q, and C = −qb are now integers. This calculator automatically applies that clearing step whenever the input coordinates are integers, so the displayed standard form always uses integer coefficients for integer inputs.

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