What are the different forms of equations of lines? Definitions and examples are presented.
The equations of lines can be written in several standard forms.
\[ y = m x + b \]
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 \( m = -2 \) and y-intercept \( (0, 3) \) is written as follows: \[ y = -2x + 3 \]
\[ y - y_1 = m (x - x_1) \]
The point-slope form is useful if the slope \( m \) and a specific 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 a slope equal to \( -3 \) may be written as follows: \[ y - 7 = -3 (x - 5) \]
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: The slope of a vertical line is undefined.
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 equation can also be written as: \[ y = 0x - 5 \]
The general equation of a line can be written as: \[ a x + b y = c \] where \( a \), \( b \), and \( c \) are constants.
Example 5: General equation of a line: \[ 2x - 5y = 8 \]