Solving Equations with Fractions

Basics of Solving Fractional Equations

• To solve an equation with fractions, eliminate the fractions by multiplying both sides by the denominator.
• Example:
  • Equation: (\frac{2x}{3} = 8)
  • Multiply by 3:
    • Left side: (\frac{2x}{3} \times 3 = 2x)
    • Right side: (8 \times 3 = 24)
  • Simplification:
    • Divide both sides by 2:
      • (2x = 24) becomes (x = 12)

Alternative Method Using Reciprocals

• Multiply by the reciprocal of the fraction.
• Example:
  • Reciprocal is (\frac{3}{2})
  • Cancel terms:
    • (\frac{2x}{3} \times \frac{3}{2} = x)
  • Compute right side:
    • (8 \times \frac{3}{2})
    • Method 1: Multiply then divide ((8 \times 3 = 24; 24 \div 2 = 12))
    • Method 2: Divide then multiply ((8 \div 2 = 4; 4 \times 3 = 12))

Solving with Larger Numbers

• Prefer dividing first to manage larger numbers easier.

Example with Added Constant

• Equation: (\frac{5x}{8} + 4 = 14)
• Steps:
  • Subtract 4 from both sides:
    • (\frac{5x}{8} = 10)
  • Multiply both sides by 8:
    • (5x = 80)
  • Divide by 5:
    • (x = 16)

Handling Equations with Multiple Fractions

• Clear fractions by multiplying by a common denominator.
• Example:
  • Equation: (\frac{1}{3} + 2x = \frac{2}{5})
  • Common denominator: 15
  • Distribute 15:
    • (15 \times \frac{1}{3} = 5)
    • (15 \times 2x = 30x)
    • (15 \times \frac{2}{5} = 6)
  • Simplify and solve:
    • Subtract 5: (30x = 1)
    • Divide by 30: (x = \frac{1}{30})

Example with Mixed Terms

• Equation: (\frac{x}{2} + 5 = \frac{1}{4})
• Use smallest common denominator (4)
• Multiply everything by 4:
  • (4 \times \frac{x}{2} = 2x)
  • (4 \times 5 = 20)
  • (\frac{1}{4} \times 4 = 1)
• Simplify:
  • (2x + 20 = 1)
• Solve:
  • Subtract 20: (2x = -19)
  • Divide by 2: (x = -\frac{19}{2})