Equations of Lines in Different Forms

What are the different forms of equations of lines? Definitions and examples are presented.

The equations of lines are of the following forms:

1 - Slope intercept form

\[ y = m x + b \]
The slope intercept form is useful if the slope \( m \) and the y intercept \( (0, b) \) of the line are known.
Example 1: The equation of a line with slope \( - 2 \) and y intercept \( (0 , 3) \) is written as follows:
\[ y = - 2 x + 3 \]

2 - Point Slope Form

\[ y - y_1 = m (x - x_1) \]
The point slope form is useful if the slope \( m \) and a point \( (x_1 , y_1) \) through which the line passes are known.
Example 2: The equation of a line that passes through the point \( (5 , 7) \) and has slope equal to \( - 3 \) may be written as follows:
\[ y - 7 = - 3 (x - 7) \]

3 - Equation of a Vertical Line

The equation of a vertical line has the form \[ x = k \] where k is a constant.
Example 3: The equation of a vertical line that passes through the point \( (-2 , -5) \) may be written as follows:
\[ x = - 2 \] Note that the slope of a vertical line is undefined.

4 - Equation of a Horizontal Line

The equation of a horizontal line has the form \[ y = k \] where \( k \) is a constant.
Example 4: The equation of a horizontal line that passes through the point \( (-2 , -5) \) may be written as follows:
\[ y = - 5 \] Its slope is equal to zero because the above equation may be written as : \( y = 0 \; x - 5 \)

5 - General Equation of a Line