Clearing Denominators

Definition

Clearing denominators is a fundamental algebraic technique used to simplify equations with fractions or decimal coefficients. It involves eliminating the denominators by multiplying both sides of the equation by a common denominator, effectively converting the fractional terms into whole numbers.

5 Must Know Facts For Your Next Test

1. Clearing denominators is a crucial step in solving equations with fractional or decimal coefficients, as it allows for easier manipulation and simplification of the equation.
2. The process of clearing denominators typically involves finding the least common denominator (LCD) of all the denominators in the equation and then multiplying both sides of the equation by the LCD.
3. Clearing denominators can be achieved through the cross-multiplication method, where each term is multiplied by the product of all the denominators.
4. Equations with cleared denominators are considered equivalent to the original equation, meaning they have the same solutions.
5. Properly clearing denominators is essential for accurately solving equations and ensuring the validity of the final solution.

Review Questions

• Explain the purpose of clearing denominators when solving equations with fractional or decimal coefficients.
  • The purpose of clearing denominators is to eliminate the fractions or decimal coefficients in an equation, making it easier to manipulate and solve. By multiplying both sides of the equation by a common denominator, typically the least common denominator (LCD), the fractional terms are converted into whole numbers. This simplifies the equation and allows for more straightforward algebraic operations, such as combining like terms and isolating the variable, ultimately leading to a more accurate solution.
• Describe the cross-multiplication method for clearing denominators and explain how it works.
  • The cross-multiplication method for clearing denominators involves multiplying each term in the equation by the product of all the denominators. This effectively eliminates the denominators, as the product of the denominators will cancel out the individual denominators in each term. The resulting equation will have whole number coefficients, making it easier to solve. The cross-multiplication method ensures that the new equation is equivalent to the original equation, meaning it has the same solutions, but with a simpler, denominator-free form.
• Analyze the importance of properly clearing denominators when solving equations and discuss the potential consequences of not doing so accurately.
  • Properly clearing denominators is crucial when solving equations with fractional or decimal coefficients, as any mistakes in this process can lead to incorrect solutions. If the denominators are not cleared correctly, the resulting equation will not be equivalent to the original, and any further algebraic manipulations will be based on an invalid equation. This can result in solutions that do not satisfy the original equation or, in some cases, may even produce nonsensical answers. Clearing denominators accurately is a fundamental skill that must be mastered to ensure the validity and reliability of the final solution to an equation with fractional or decimal coefficients.

Related terms

Least Common Denominator (LCD):

The smallest positive integer that is divisible by all the denominators in an equation, used to clear the denominators.

Cross-Multiplication: A method of clearing denominators by multiplying both sides of the equation by the product of all the denominators.

Equivalent Equations: Equations that have the same solutions, even though they may appear different due to the clearing of denominators.