Solving equations with fractions and decimals can be tricky, but it's a crucial skill in algebra. By learning to clear denominators and handle decimal coefficients, you'll be able to tackle more complex problems with confidence.

Converting between fractions and decimals is another key technique. This flexibility allows you to choose the most efficient approach for each problem, making your equation-solving process smoother and more effective.

Solving Equations with Fractions and Decimals

Clearing denominators in equations

• Multiply both sides of the equation by the least common denominator (LCD) of all fractions to clear denominators
  • List the denominators and find the least common multiple to determine the LCD (12 is the LCD of 2, 3, and 4)
  • Multiply each term on both sides of the equation by the LCD to eliminate fractions ($2x/3 + 1/4 = 5/6$ becomes $8x + 3 = 10$)
  • Multiply out parentheses and combine like terms to simplify the resulting equation ($2(3x - 1) = 4x + 6$ becomes $6x - 2 = 4x + 6$)
• Use standard methods to solve the equation after clearing denominators
  • Isolate the variable term on one side of the equation by adding or subtracting terms ($6x - 2 = 4x + 6$ becomes $2x - 2 = 6$)
  • Perform the same operation on both sides to maintain equality (adding 2 to both sides gives $2x = 8$)
  • Simplify each side of the equation to find the solution (dividing both sides by 2 gives $x = 4$)
• Finding the common denominator is crucial when working with rational equations

Strategies for decimal coefficients

• Eliminate decimals by multiplying both sides of the equation by a power of 10
  • Determine the smallest power of 10 that will make all coefficients whole numbers (multiplying by 100 if the smallest decimal place is hundredths)
  • Multiply each term on both sides of the equation by the chosen power of 10 ($0.3x - 0.7 = 1.2$ becomes $30x - 70 = 120$)
  • Perform the multiplication to simplify the resulting equation ($0.2(4x + 1) = 0.6x$ becomes $0.8x + 0.2 = 0.6x$)
• Use standard methods to solve the equation after eliminating decimals
  • Isolate the variable term on one side of the equation by adding or subtracting terms ($0.8x + 0.2 = 0.6x$ becomes $0.2x + 0.2 = 0$)
  • Perform the same operation on both sides to maintain equality (subtracting 0.2 from both sides gives $0.2x = -0.2$)
  • Simplify each side of the equation to find the solution (dividing both sides by 0.2 gives $x = -1$)

Fraction-decimal conversions for equations

• Divide the numerator by the denominator to convert fractions to decimals
  • Use long division or a calculator to divide the numerator by the denominator (3/4 = 0.75)
  • Express the result as a decimal, rounding if necessary (5/3 ≈ 1.67 rounded to two decimal places)
• Write the decimal as a fraction over a power of 10 to convert decimals to fractions
  • Write the decimal as a whole number over a power of 10 based on the number of decimal places (0.6 = 6/10)
  • Divide both the numerator and denominator by their greatest common factor (GCF) to simplify the resulting fraction (0.75 = 75/100 = 3/4)
• Choose the form (fraction or decimal) that simplifies the equation-solving process
  • Convert one form to the other to have a consistent representation if the equation has both fractions and decimals ($2x/5 + 0.3 = 3/4$ becomes $0.4x + 0.3 = 0.75$)
  • Select the form that leads to simpler calculations or avoids complicated fractions or lengthy decimals (using decimals for $0.2x + 0.5 = 1.3$ instead of fractions)

Working with Algebraic Fractions

• Algebraic fractions are fractions where the numerator, denominator, or both contain variables
• Simplify algebraic fractions by factoring and canceling common factors
• When solving equations with algebraic fractions, use the reciprocal to eliminate fractions
• Apply the same principles as solving equations with numeric fractions, but be cautious of potential undefined values