Writing Equations of Lines: Parallel and Perpendicular

Introduction

• Focus on writing equations of lines that are parallel or perpendicular to a given line.
• Use the point-slope form and slope-intercept form.

Parallel Lines

• Key Concept: Parallel lines have the same slope.

• Example 1: Given the line $y = 2x + 1$, write the equation of a line parallel to it passing through (1, 5).

  • Slope (m) is 2.
  • Use the slope-intercept form: $y = mx + b$.
  • Substitute $x = 1$, $y = 5$: $5 = 2(1) + b \Rightarrow b = 3$.
  • Equation: $y = 2x + 3$.

• Example 2: Given $y = 3x - 4$, find the parallel line through (6, 4).

  • Slope (m) is 3.
  • Substitute $x = 6$, $y = 4$: $4 = 3(6) + b \Rightarrow b = -14$.
  • Equation: $y = 3x - 14$.

Using Point-Slope Form

• Example 3: Line $y = \frac{3}{2}x + 3$, point (4, -3).
  • Slope (m) is $\frac{3}{2}$.
  • Point-slope form: $y - y_1 = m(x - x_1)$.
  • Substitute $x_1 = 4$, $y_1 = -3$: $y + 3 = \frac{3}{2}(x - 4)$.
  • Distribute: $y + 3 = \frac{3}{2}x - 6$.
  • Convert to slope-intercept: $y = \frac{3}{2}x - 9$.

Perpendicular Lines

• Key Concept: Perpendicular lines have slopes that are negative reciprocals.

• Example 4: Given $y = 5x - 3$, find the perpendicular line through (2, 1).

  • Original slope is 5.
  • Perpendicular slope: $-\frac{1}{5}$.
  • Point-slope form: $y - y_1 = m(x - x_1)$.
  • Substitute $x_1 = 2$, $y_1 = 1$: $y - 1 = -\frac{1}{5}(x - 2)$.
  • Distribute: $y = -\frac{1}{5}x + \frac{7}{5}$.

• Example 5: Given $y = \frac{3}{4}x - 1$, perpendicular line through (8, -3).

  • Perpendicular slope: $-\frac{4}{3}$.
  • Substitute $x = 8$, $y = -3$: $-3 = -\frac{4}{3}(8) + b$.
  • Solve for $b$: $b = \frac{23}{3}$.
  • Equation: $y = -\frac{4}{3}x + \frac{23}{3}$.

Final Example

• Example 6: $y = -\frac{2}{3}x + 4$, find perpendicular line through (6, -2).
  • Perpendicular slope: $\frac{3}{2}$.
  • Substitute $x = 6$, $y = -2$: $-2 = \frac{3}{2}(6) + b$.
  • Solve for $b$: $b = -11$.
  • Equation: $y = \frac{3}{2}x - 13$.

Conclusion

• Practicing these steps with various examples helps solidify understanding of writing equations for parallel and perpendicular lines.