All Three Forms of the Equation of a Line

INTRODUCTION

A straight line is one of the basic structures in mathematics, which can be represented in various ways. The three main forms of the equation of a line are the explicit form (slope-intercept form), the implicit form (general form), and the intercept form. Each of these forms has its role and is useful in different mathematical contexts.

1. EXPLICIT FORM (SLOPE-INTERCEPT FORM)

The explicit form of the equation of a line is the most commonly used and is written as y = kx + n, where:

• k is the slope (or gradient), which tells how steeply the line is inclined.
• n is the y-intercept, which shows where the line crosses the y-axis (the value of y when x=0).
• x and y are the coordinates of points on the line.

2. IMPLICIT FORM (GENERAL FORM)

The implicit form of the equation of a line is written as ax + by + c = 0, where:

• a, b, and c are constants (with 'a' and 'b' not both zero).
• x and y are variables representing coordinates on the line.

3. INTERCEPT FORM

The intercept form of the equation of a line is given as x/m + y/n = 1, where:

• m is the x-intercept (the value of x where the line crosses the x-axis, so y=0).
• n is the y-intercept (the value of y where the line crosses the y-axis, so x=0).
  (This form is valid when m ≠ 0 and n ≠ 0).