Linear Equation to Standard Form Calculator

Inputs

Input Mode

Example: y = mx + b

Options

Force A ≥ 0

Reduce by GCD

Clear fractions to integers

Ax + By = C

A: –, B: –, C: –

Share: (generate)

Compute to see detailed algebra here.

#	Mode	Inputs	A	B	C	Use

Input Form	Given	Standard Form Mapping	Notes
Slope–Intercept	y = m x + b	A = m, B = −1, C = −b	Then clear denominators, reduce, and enforce A ≥ 0.
Point–Slope	y − y₁ = m(x − x₁)	A = m, B = −1, C = m x₁ − y₁	Use exact fractions for m, x₁, y₁ if entered.
Two Points	(x₁,y₁), (x₂,y₂)	m = (y₂−y₁)/(x₂−x₁), then as Slope–Intercept	Vertical if x₁ = x₂ → A = 1, B = 0, C = x₁.
Coefficients	a x + b y = c	A = a, B = b, C = c	We can still reduce and flip sign if needed.
Intercepts	x/a + y/b = 1	A = b, B = a, C = a b	Multiply by ab to eliminate denominators first.

Condition	Detection	Standard Form Output (A,B,C)	Example
Horizontal line	m = 0	A = 0, B ≠ 0, C = b	y = 5 → 0·x + 1·y = 5
Vertical line	x₁ = x₂ in two-point, or undefined slope	A = 1, B = 0, C = x₁	x = −3 → 1·x + 0·y = −3
All-zero coefficients	A = B = C = 0 after parse	Invalid; prompt for new inputs	Empty fields → show validation message
Negative A after reduction	A < 0 with option “Force A ≥ 0” enabled	Multiply triplet by −1	−2x − 4y = −6 → 2x + 4y = 6
Fractional coefficients	Any denominators present	Scale by LCM of denominators; then reduce by gcd	½x − ⅓y = 2 → 3x − 2y = 12

Slope–Intercept y = m x + b ⇒ m x − y = −b ⇒ A=m, B=−1, C=−b.
Point–Slope y − y₁ = m (x − x₁) ⇒ m x − y = m x₁ − y₁ ⇒ A=m, B=−1, C=m x₁ − y₁.
Two-Point slope m = (y₂ − y₁)/(x₂ − x₁) then use Slope–Intercept result.
Coefficients given a x + b y = c ⇒ A=a, B=b, C=c.
Intercepts x/a + y/b = 1 ⇒ multiply by ab: b x + a y = ab ⇒ A=b, B=a, C=ab.
Then clear denominators, reduce by gcd(|A|,|B|,|C|), enforce A ≥ 0.

It’s a common convention so the representation is unique. If A is negative we multiply both sides by −1, preserving the same line.

We convert decimals to exact rational numbers by using powers of ten, then simplify with the greatest common divisor before clearing denominators.

Horizontal lines have B ≠ 0 and A = 0; vertical lines have A ≠ 0 and B = 0. The tool handles both and keeps integer coefficients.

That’s a vertical line. We directly return x = x₁ as Ax + By = C with A = 1, B = 0, C = x₁ after clearing denominators.

Yes. Use forms like 2/3, -7/5, or mixed decimals. The parser accepts integers, decimals, and rational strings with optional signs.

Dividing A, B, and C by their greatest common divisor produces the simplest integer triplet representing the same line.

No. Decimals are converted to exact fractions first, so clearing denominators is exact rather than approximate.