Equation of a line

Introduction

An equation of a line is a mathematical statement that describes all the points that lie on that particular line in a coordinate plane. It provides a relationship between the x-coordinate (abscissa) and y-coordinate (ordinate) of every point on the line.

In simpler terms, if we plug any x-value into the equation, it gives us the corresponding y-value for a point on the line, and vice versa. The equation of a line helps us understand the line's slope (its steepness and direction) and intercepts (where it crosses the x-axis and y-axis).

Various forms of the equation of a line

  1. Slope-Intercept Form

    The slope-intercept form of a line's equation is given by: $$ y = mx + b $$ where,

    • 𝑦 is the dependent variable (ordinate)
    • 𝑥 is the independent variable (abscissa)
    • 𝑚 is the slope of the line
    • 𝑏 is the y-intercept (the point where the line crosses the y-axis)
    Example:
    Consider the equation $$ y = 2x + 3 $$ In this equation, the slope 𝑚 is 2, and the y-intercept 𝑏 is 3. This means that for every 1 unit increase in 𝑥 , 𝑦 increases by 2 units, and the line crosses the y-axis at the point (0, 3).

  2. Point-slope form

    The point-slope form of a line's equation is given by: $$ 𝑦 − 𝑦 1 = 𝑚 ( 𝑥 − 𝑥 1 ) $$ where: ( 𝑥 1 , 𝑦 1 ) is a point on the line 𝑚 is the slope of the line
    Example:
    Consider the point (4, 5) and a slope 𝑚 of 3. Using the point-slope form, the equation of the line becomes: $$ 𝑦 − 5 = 3 ( 𝑥 − 4 ) $$ This equation represents a line that passes through the point (4, 5) with a slope of 3.
  3. Two-Point Form

    The two-point form of a line's equation is used when two points on the line are known. It is given by: $$ \frac{ 𝑦 − 𝑦_ 1 }{ 𝑦_ 2 − 𝑦_ 1 } = \frac{ 𝑥 − 𝑥_ 1 }{ 𝑥_ 2 − 𝑥_ 1 } $$ where ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 ) are two points on the line.
    Example:
    Consider the points (2, 3) and (5, 7). Using the two-point form, the equation of the line becomes: $$ \frac{ 𝑦 − 3 }{ 7 − 3 } = \frac{ 𝑥 − 2 }{ 5 − 2 } $$ Simplifying this equation, we get: $$ \frac{ 𝑦 − 3 }{ 4 } = \frac{ 𝑥 − 2 }{ 3 } $$ Cross-multiplying, we get the linear equation: $$ 3(y - 3) = 4(x - 2) $$ $$ 3y - 9 = 4x - 8 $$ $$ 4x - 3y = -1 $$
  4. Doduble Intercept Form

    The intercept form of a line's equation is given by: $$ \frac {𝑥} {𝑎} + \frac {𝑦} {𝑏} = 1 $$ where 𝑎 and 𝑏 are the x-intercept and y-intercept, respectively.
    Example:
    Consider the x-intercept 𝑎 = 4 and y-intercept 𝑏 = 2 . Using the double intercept form, the equation of the line becomes: $$ \frac {𝑥} {4} + \frac {𝑦} {2} = 1 $$ Multiplying through by 4 to clear the denominator, we get: $$ x + 2y = 4 $$
  5. Standard Form

    The standard form of a line's equation is given by: $$ 𝐴 𝑥 + 𝐵 𝑦 = 𝐶 $$ where: 𝐴 , 𝐵 , and 𝐶 are integers (with 𝐴 and 𝐵 not both zero)
    Example:
    Consider the equation $$ 3 𝑥 + 4 𝑦 = 12 . $$ In this form, it is often useful for solving systems of linear equations and performing algebraic manipulations.