Line Formula

The Formula

y = mx + b (slope-intercept form in the coordinate plane)

When to use: A perfectly straight edge that goes on forever in both directions.

Quick Example

The line through points A and B extends past both, forever.

Notation

\overleftrightarrow{AB} denotes the line through A and B; \overline{AB} is a segment; \overrightarrow{AB} is a ray from A through B

What This Formula Means

A perfectly straight path extending infinitely in both directions through two distinct points, with no thickness.

A perfectly straight edge that goes on forever in both directions.

Formal View

\ell_{A,B} = \{A + t(B - A) : t \in \mathbb{R}\} for distinct points A, B \in \mathbb{R}^n; in \mathbb{R}^2: \{(x,y) : ax + by = c\} for some (a,b) \neq (0,0)

Worked Examples

Example 1

easy
Write the equation of a line with slope m = 2 and y-intercept b = -3.

Solution

  1. 1
    Step 1: The slope-intercept form of a line is y = mx + b.
  2. 2
    Step 2: Substitute m = 2 and b = -3.
  3. 3
    Step 3: The equation is y = 2x - 3.

Answer

y = 2x - 3
In y = mx + b, m is the slope (rise over run) and b is where the line crosses the y-axis. A line extends infinitely in both directions — every point on it satisfies this equation.

Example 2

medium
Find the slope of the line passing through points (1, 3) and (4, 9).

Common Mistakes

  • Drawing a line with endpoints (that's a segment) — a line extends infinitely in both directions
  • Confusing a line (infinite both ways) with a ray (infinite one way) or a segment (finite)
  • Thinking two lines must intersect — parallel lines in the same plane never meet

Why This Formula Matters

Lines define directions, boundaries, and linear relationships.

Frequently Asked Questions

What is the Line formula?

A perfectly straight path extending infinitely in both directions through two distinct points, with no thickness.

How do you use the Line formula?

A perfectly straight edge that goes on forever in both directions.

What do the symbols mean in the Line formula?

\overleftrightarrow{AB} denotes the line through A and B; \overline{AB} is a segment; \overrightarrow{AB} is a ray from A through B

Why is the Line formula important in Math?

Lines define directions, boundaries, and linear relationships.

What do students get wrong about Line?

Line vs segment vs ray: line goes forever; segment has endpoints; ray has one endpoint.

What should I learn before the Line formula?

Before studying the Line formula, you should understand: point.

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Related Concepts

PointPlane

Related Formulas

Plane Formula