Solving Equations with Fractions

Strategy Overview

• Multiply both sides of an equation by a common denominator to eliminate fractions.
• This works because multiplying both sides by the same number retains the equation's balance.
• Common denominators help simplify equations by removing fractions.

Example Problem 1

• Fractions & Denominators:
  • Denominators: 5, 15, 3
  • Multiply by common denominator: 15
• Steps:
  1. Distribute 15 across each term:
     • (15/5) * (x - 8) = 3 * (x - 8)
     • (15/15) * 8 = 8
     • (15/3) * (-5x) = -5 * 5x
  2. Distribute and simplify:
     • 3x - 24 + 8 = -5x
  3. Combine like terms:
     • 3x - 16 = -5x
  4. Solve for x:
     • Subtract 3x from both sides: -16 = -8x
     • Divide by -8: x = 2

Example Problem 2

• Fractions & Denominators:
  • Common denominator: 10
• Steps:
  1. Multiply each term by 10:
     • (10/5) * (2x + 5x) = 2 * (2x + 5x)
     • (10/2) * (3x + 1) = 5 * (3x + 1)
     • (10/2) * (-x + 7) = 5 * (-x + 7)
  2. Distribute and simplify:
     • 4x + 10 = 15x - 5x + 35
  3. Combine like terms:
     • 4x + 10 = 10x + 40
  4. Solve for x:
     • Subtract 10x from both sides: -6x + 10 = 40
     • Subtract 10: -6x = 30
     • Divide by -6: x = -5

Example Problem 3

• Fractions & Denominators:
  • Fractions: 1/3, 1/4
  • Whole number consideration: denominator is 1
  • Common denominator: 12
• Steps:
  1. Multiply each term by 12:
     • (12/3) * x = 4x
     • (12/4) * 1 = 3
     • Multiply 7 by 12 = 84
  2. Rearrange equation:
     • 4x + 3 = 84
  3. Solve for x:
     • Subtract 3 from both sides: 4x = 81
     • Divide by 4: x = 81/4

Key Takeaways

• Use a common denominator to simplify equation solving by eliminating fractions.
• Distribute the multiplier to each term, and ensure both sides of the equation remain balanced.
• Combine like terms and solve for the variable step by step.