General Form of a Line — Definition, Formula & Examples

The general form of a line is a way of writing a linear equation as

Ax + By + C = 0

, where

A

B

, and

C

are real numbers and

A

and

B

are not both zero.

A linear equation in two variables $x$ and $y$ is said to be in general form when expressed as $Ax + By + C = 0$ , where $A, B, C \in \mathbb{R}$ and $(A, B) \neq (0, 0)$ . By convention, $A$ is typically written as a positive integer when the coefficients are integers.

Key Formula

Ax + By + C = 0

Where:

$A$ = Coefficient of x (real number)
$B$ = Coefficient of y (real number)
$C$ = Constant term (real number)
$x$ = Independent variable
$y$ = Dependent variable

How It Works

Any linear equation can be rearranged into general form by moving all terms to one side so the equation equals zero. To convert from slope-intercept form (

y = mx + b

), subtract

y

from both sides and rearrange. To convert from general form to slope-intercept form, isolate

y

y = -\frac{A}{B}x - \frac{C}{B}

(provided

B \neq 0

). The slope of the line is

-\frac{A}{B}

and the

y

-intercept is

-\frac{C}{B}

Worked Example

Problem: Write the equation y = 3x − 5 in general form.

Step 1: Subtract y from both sides to begin moving all terms to one side.

0 = 3x - y - 5

Step 2: Rewrite with terms on the left side, set equal to zero.

3x - y - 5 = 0

Step 3: Identify the coefficients: A = 3, B = −1, C = −5. Since A is positive and all coefficients are integers, this is already in standard convention.

Answer: The general form is

3x - y - 5 = 0

Why It Matters

General form is especially useful when working with systems of equations, since it keeps both variables on the same side for easy elimination. It also appears in analytic geometry when finding distances from points to lines, where the formula requires the equation in

Ax + By + C = 0

form.

Common Mistakes

Mistake: Confusing general form (Ax + By + C = 0) with standard form (Ax + By = C).

Correction: In general form, everything is on one side equaling zero. In standard form, the constant is isolated on the right side. Both are valid ways to write a line, but they differ by where C sits.