Equation of a Line

The equation of a line represents the relationship between the \( x \)- and \( y \)-coordinates of points on that line. There are several forms for writing the equation of a line, depending on the information available. Each form has its specific uses in algebra, geometry, and other applications where line equations are needed.

Forms of the Equation of a Line

Slope-Intercept Form:

If the slope \( m \) and the y-intercept \( b \) of the line are known, the equation of the line can be written in slope-intercept form:

\( y = mx + b \)

Point-Slope Form:

If a point \( (x_1, y_1) \) on the line and the slope \( m \) are known, the equation of the line can be written in point-slope form:

\( y – y_1 = m(x – x_1) \)

Standard Form:

If the equation of the line needs to be written without fractions or decimals, it can be expressed in standard form:

\( Ax + By = C \)

where \( A \), \( B \), and \( C \) are integers, and \( A \) is non-negative.

Detailed Explanation of Each Form

1 Slope-Intercept Form

The slope-intercept form, \( y = mx + b \), is useful when the slope and the y-intercept are known. Here:

  • \( m \) represents the slope of the line
  • \( b \) represents the y-intercept, the point where the line crosses the y-axis

Example 1: Writing the Equation of a Line in Slope-Intercept Form

Problem: Write the equation of a line with a slope \( m = 3 \) and y-intercept \( b = -2 \).

Solution:

  1. Use the formula \( y = mx + b \).
  2. Substitute \( m = 3 \) and \( b = -2 \): \( y = 3x – 2 \).

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

2 Point-Slope Form

The point-slope form, \( y – y_1 = m(x – x_1) \), is useful when you know a point on the line \( (x_1, y_1) \) and the slope \( m \). This form is often used to derive the line equation when the coordinates of a specific point and the slope are available.

Example 2: Writing the Equation of a Line in Point-Slope Form

Problem: Write the equation of a line that passes through the point \( (1, 4) \) and has a slope of \( m = -2 \).

Solution:

  1. Use the formula \( y – y_1 = m(x – x_1) \).
  2. Substitute \( x_1 = 1 \), \( y_1 = 4 \), and \( m = -2 \): \( y – 4 = -2(x – 1) \).
  3. Simplify: \( y – 4 = -2x + 2 \).
  4. Rewrite in slope-intercept form if needed: \( y = -2x + 6 \).

The equation of the line in point-slope form is \( y – 4 = -2(x – 1) \), and in slope-intercept form, it’s \( y = -2x + 6 \).

3 Standard Form

The standard form, \( Ax + By = C \), is often used when you want to express the line equation without fractions or decimals. In this form, \( A \), \( B \), and \( C \) are integers, and \( A \) is non-negative.

Example 3: Converting an Equation to Standard Form

Problem: Convert the equation \( y = \dfrac{3}{4}x – 5 \) to standard form.

Solution:

  1. Start with the slope-intercept form: \( y = \dfrac{3}{4}x – 5 \).
  2. Multiply both sides by 4 to eliminate the fraction: \( 4y = 3x – 20 \).
  3. Rearrange terms to get \( 3x – 4y = 20 \), the standard form.

The equation of the line in standard form is \( 3x – 4y = 20 \).

These three forms—slope-intercept, point-slope, and standard—each provide different ways to express the equation of a line, depending on the information available and the format required.