What is Standard Form of a Line

The standard form of a line equation is useful for expressing a line in terms of integer coefficients, which is particularly helpful for identifying intercepts and comparing different lines. This form is widely used in algebra and geometry. The standard form of a line is expressed as:

\( Ax + By = C \)

In this formula:

  • \( A \), \( B \), and \( C \) are integers (whole numbers)
  • \( A \) should be non-negative (if possible)
  • \( x \) and \( y \) are the variables representing points on the line

The standard form is useful because it avoids fractions and makes it easy to calculate the x- and y-intercepts of the line.

How to Use the Standard Form

The standard form, \( Ax + By = C \), is often used to:

  • Find the x-intercept by setting \( y = 0 \) and solving for \( x \)
  • Find the y-intercept by setting \( x = 0 \) and solving for \( y \)

If needed, the standard form can be converted to slope-intercept form, \( y = mx + b \), by isolating \( y \) on one side of the equation.

Converting Slope-Intercept Form to Standard Form

If the line equation is given in slope-intercept form, \( y = mx + b \), we can convert it to standard form by rearranging terms and eliminating any fractions:

  • Start with \( y = mx + b \).
  • Rearrange terms to get \( mx – y = -b \) (if \( m \) and \( b \) are integers, no further steps are required).
  • If needed, multiply all terms by a common denominator to clear fractions.

Example 1: Converting a Slope-Intercept Equation to Standard Form

Problem: Convert the equation \( y = \dfrac{3}{2}x – 5 \) to standard form.

Solution:

  1. Start with the slope-intercept form: \( y = \dfrac{3}{2}x – 5 \).
  2. Multiply all terms by 2 to eliminate the fraction: \( 2y = 3x – 10 \).
  3. Rearrange to get \( 3x – 2y = 10 \).

The equation of the line in standard form is \( 3x – 2y = 10 \).

Example 2: Finding Intercepts from Standard Form

Problem: Find the x- and y-intercepts of the line with the equation \( 4x + 5y = 20 \).

Solution:

To find the x-intercept, set \( y = 0 \):

\( 4x + 5(0) = 20 \Rightarrow x = 5 \)

The x-intercept is \( (5, 0) \).

To find the y-intercept, set \( x = 0 \):

\( 4(0) + 5y = 20 \Rightarrow y = 4 \)

The y-intercept is \( (0, 4) \).

The x-intercept is \( (5, 0) \), and the y-intercept is \( (0, 4) \).

The standard form \( Ax + By = C \) provides a flexible way to work with line equations, especially for calculating intercepts or rearranging equations for specific applications.