Lecture on Equations of Lines

Key Concepts

• Equation of a Line: Typically presented in slope-intercept form as ( y = mx + b )
  • ( m ) = Slope of the line
  • ( b ) = Y-intercept of the line

Perpendicular Lines

• Definition: Perpendicular lines have slopes that are opposite reciprocals.
  • For a slope ( m ), the perpendicular slope is (-1/m).
  • If the original slope is negative, the perpendicular slope will be positive and vice versa.

Example

1. Given Slope: (-2)

   • Perpendicular slope: (1/2)

2. Forming a New Equation:

   • Start with ( y = \frac{1}{2}x + b )

3. Finding the Y-intercept ((b)):

   • Use a point ((x, y)) to solve for ( b ).
   • Example Point: ((-3, -2))

4. Calculation:

   • Plug in values:
     • ( -2 = \frac{1}{2}(-3) + b )
   • Simplify:
     • ( -2 = -\frac{3}{2} + b )
   • Add ( \frac{3}{2} ) to both sides:
     • ( -2 + \frac{3}{2} = b )
     • Simplify ( -2 + 1.5 = -0.5 )
   • Therefore, ( b = -\frac{1}{2} )

5. Complete Equation:

   • The new equation of the line: ( y = \frac{1}{2}x - \frac{1}{2} )

Summary

• To find the equation of a line perpendicular to a given line:
  • Determine the reciprocal and opposite of the original slope for the perpendicular slope.
  • Use a given point to solve for the new y-intercept ( b ).
  • Form the equation with the new slope and y-intercept.