Different Forms Of The Equation Of Line

A line is a set of points that goes on forever in both directions. In mathematics, we can represent a line using an equation.
In this article, we'll explore the different forms of the equation of a line and how to work with them. We'll look at the key components and learn how to use them to describe and analyze lines.

Introduction to Equation Of Line

Lines are a big part of our everyday life, appearing in many different forms around us. By definition, a line is a collection of points that goes on forever in both directions.

In geometry, a line extends in a straight, unbroken path with no curves and has no width or thickness.

A line is considered straight if it extends in a straight path without any bends. A section of a line with two endpoints is called a line segment.

The mathematical way to describe a line is called the “Equation of a Line.” It's written as y = mx + c, where m is the slope and c is the y-intercept.

The slope of a line shows how steep the line is, and the y-intercept is where the line crosses the y-axis.

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Different Forms of the Equation of a Line

There are several ways to write the equation of a line:

1. Normal Form
2. Intercept Form
3. Slope-Intercept Form
4. Two-Point Form
5. Point-Slope Form

Normal Form

Also known as Parametric Normal Form or Perpendicular Distance Form, this line is perpendicular to another line passing through the origin. It can be derived from the slope-intercept form or point-slope form.

Example: For a line AB perpendicular to the line passing through the origin (line C),

• Coordinates of B: (C cos θ, C sin θ)
• Slope of line AB: tan θ
• Slope of line C: -cot θ = cos θ / sin θ

So, the equation of line C is:

y - C sin θ = - (cos θ / sin θ) (x - C cos θ)
x cos θ + y sin θ = C

Intercept Form

For a line with x-intercept p and y-intercept q, touching the x-axis at (p,0) and y-axis at (0,q), the equation is:

x/p + y/q = 1

Example: If p = 5 and q = 6, then:

x/5 + y/6 = 1
6x + 5y = 30

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Slope-Intercept Form

Represented by y = mx + c, where:

• m is the slope
• c is the y-intercept

Example: For a line with slope m and y-intercept p, the equation is:

y = mx + p

Two-Point Form

This form finds the equation of a line given two points (x1, y1) and (x2, y2). The slope between the points is:

slope = (y2 - y1) / (x2 - x1)

Thus, the equation is:

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

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Point-Slope Form

Given a point (x1, y1) and slope m, the equation is:

y - y1 = m (x - x1)

Example: For a line with slope m = 3 passing through (4, 5), the equation is:

y - 5 = 3 (x - 4)
y = 3x - 7
3x - y - 7 = 0