Calculating the Slope of a Line Between Two Points

Introduction

• Focus on finding the slope between two given points
• Use the formula: Slope ( m = \frac{y_2 - y_1}{x_2 - x_1} )

Example Calculations

Example 1

• Points: (2,5) and (6,13)
  • ( x_1 = 2, y_1 = 5, x_2 = 6, y_2 = 13 )
  • Slope ( m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2 )

Example 2

• Points: (3,-4) and (-5,2)
  • ( x_1 = 3, y_1 = -4, x_2 = -5, y_2 = 2 )
  • Simplification: ( 2 - (-4) = 6 ) and ( -5 - 3 = -8 )
  • Reduced: ( \frac{6}{-8} = \frac{-3}{4} )

Example 3

• Points: (3,6) and (5,6)
  • Slope = 0
  • Implication: Horizontal line

Example 4

• Points: (4,1) and (4,-8)
  • Slope is undefined (vertical line)

Example with Fractions

• Points with fractions: ((-\frac{1}{3}, \frac{1}{5})) and (3,1)
  • Use least common multiple to clear fractions
  • Slope ( \frac{9}{100} )

Advanced Examples with Variables

Find the Missing Coordinate

• Given points involving variables and a slope

Example 1

• Points: (3,y) and (2,5), slope = ( \frac{3}{4} )
  • Solve for ( y ) using cross-multiplication
  • Result: ( y = \frac{23}{4} )

Example 2

• Points: ((-\frac{1}{2}, \frac{1}{3})) and (x, (\frac{1}{4})), slope = ( -\frac{5}{3} )
  • Solve for ( x ) using similar steps
  • Result: ( x = -\frac{9}{20} )

Understanding Slope in Equations

Slope-Intercept Form

• Form: ( y = mx + b )
• Example: Given ( y = 2x + 5 ), slope ( m = 2 )

Reverse Order

• Example: ( y = 5 - \frac{2}{3}x )
  • Slope ( m = -\frac{2}{3} )
  • Y-intercept = 5

Standard Form to Slope-Intercept Form

• Convert by isolating ( y )
  • Example: (-2y = -4x + 8 )
  • Rearrange: ( y = 2x - 4 ), slope ( m = 2 )

Summary

• Calculating slopes involves understanding the change in y over the change in x.
• Special lines: Horizontal (slope = 0), Vertical (slope undefined)
• Applications involve fractions, variables, and converting between equation forms.