Solving Linear Equations with Fractions

Introduction

• The video discusses solving linear equations that include fractions.
• Aim: Make equations easier by eliminating fractions.

Steps to Solve Equations

1. Identify the Least Common Denominator (LCD)

   • Find the smallest number divisible by all denominators in the equation.
   • Example: In the equation 4x/5 = 5x/3 - 13, denominators are 3 and 5, LCD is 15.

2. Multiply Each Term by the LCD

   • Multiply both sides by the LCD to eliminate fractions.
   • Example transformation:
     • 15 * (4x/5) = 15 * (5x/3 - 13)

3. Apply the Distributive Property

   • Distribute the LCD across terms:
     • 15 * (4x/5) = 15 * (5x/3) - 15 * 13

4. Simplify and Solve

   • Divide out common factors.
   • Example:
     • 15/5 = 3 and 15/3 = 5
   • Complete multiplication:
     • 3 * 4x = 12x and 5 * 5x = 25x, 15 * 13 = 195
     • Results in: 12x = 25x - 195

5. Isolate the Variable

   • Rearrange terms to isolate x:
     • 12x - 25x = -195
     • Simplify: -13x = -195
   • Divide both sides by -13:
     • x = 15

6. Verify Solution

   • Substitute x back into the original equation to verify.
   • Confirm: 12 = 12, solution set is {15}.

Application: Responding to Negative Life Events

Background

• Two line graphs (low humor and high humor groups) represent responses to negative life events.
• Graph axes:
  • Horizontal: Intensity from 1 (low) to 10 (high).
  • Vertical: Depression level from 2 to 18.

Analysis of Graphs

• Low humor group has steeper and higher depression response.
• Formulas represent depression levels:
  • Low Humor: D = (10/9)x + 53/9
  • High Humor: D = (1/9)x + 26/9

Solving for Intensity

• Given: High humor group depression is 3.5 (or 7/2).

• Substitute into the high humor equation and solve for x.

  1. Replace D with 7/2.
  2. Multiply by the LCD (18):
     • 18 * (7/2) = 18 * ((1/9)x + 26/9)
  3. Apply distributive property:
     • 18 * (7/2) = 18 * (1/9)x + 18 * (26/9)
  4. Simplify and solve:
     • 63 = 2x + 52
     • Subtract 52: 63 - 52 = 2x
     • Simplify: 11 = 2x
     • Divide by 2: x = 11/2

• Conclusion: Intensity for a 3.5 depression level is 11/2 or 5.5, shown on the high humor graph.*