Lecture Notes: Equation of a Line - Slope-Intercept and Point-Slope Form

Introduction

• Focus on linear equations of the form ( y = mx + b )
• Known as the slope-intercept form
  • ( m ): slope of the line
  • ( b ): y-intercept (point where the line crosses the y-axis)
• Advantage: Simple, but requires exact y-intercept
• Alternative: Use when y-intercept is unknown, but another point is known

Slope-Intercept Form

• Equation: ( y = mx + b )
• Constants:
  • m: Slope
  • b: Y-intercept
• Useful for determining the equation if y-intercept is known

Point-Slope Form

• Used when it’s difficult to determine y-intercept
• Equation derived from the slope formula: [ \frac{y_2 - y_1}{x_2 - x_1} = m ]
• Requires:
  • Known slope ( m )
  • Coordinates of a point ((x_1, y_1)) on the line
• Equation: ( y - y_1 = m(x - x_1) )
• Easier to use with known slope and any point on the line

Example 1:

• Given: Slope ( m = \frac{3}{5} )
• Known point: ( (4,3) )
• Equation: ( y - 3 = \frac{3}{5}(x - 4) )
• Can be rearranged to have ( y ) on one side
  • Simplifies to ( y = \frac{3}{5}(x - 4) + 3 )

Example 2:

• Given: Slope ( m = -\frac{2}{3} )
• Known point: ( (-7,3) )
  • Simplified notation: ( y - 3 = -\frac{2}{3}(x + 7) )
• Finding the y-intercept:
  • Set ( x = 0 ) to find y-value
  • Calculated ( y ) as ( -\frac{5}{3} )
• Equation can be written in slope-intercept form:
  • ( y = -\frac{2}{3}x - \frac{5}{3} )

Equivalence of Forms

• Both point-slope and slope-intercept forms describe the same line
• Conversion:
  • Use distribution and arithmetic to transform point-slope to slope-intercept
  • Both forms express equivalent relations between ( x ) and ( y )

Conclusion

• The form to use depends on the known information about the line
• Preview: Two-point form of a line equation in the next lecture