key term - Proportion Method

Definition

The proportion method is a technique used to solve rational equations by setting up a proportion between the numerators and denominators of the fractions involved. This method allows for the identification of a common factor that can be used to simplify the equation and find the solution.

5 Must Know Facts For Your Next Test

1. The proportion method involves setting up a proportion between the numerators and denominators of the rational expressions in the equation.
2. Cross-multiplication is a key step in the proportion method, where the means of the proportion are set equal to the product of the extremes.
3. Simplifying the resulting equation by finding a common factor is an important part of the proportion method.
4. The proportion method is particularly useful when the rational equation contains variables in both the numerator and denominator.
5. Finding the least common denominator (LCD) of the fractions in the rational equation is often necessary when applying the proportion method.

Review Questions

• Explain the purpose and key steps of the proportion method when solving rational equations.
  • The proportion method is used to solve rational equations by setting up a proportion between the numerators and denominators of the fractions involved. The key steps are: 1) Set up a proportion by equating the ratios of the numerators and denominators, 2) Cross-multiply the terms in the proportion to create a new equation, 3) Simplify the resulting equation by finding a common factor, and 4) Solve the simplified equation to find the solution(s) to the original rational equation.
• Describe how the concept of the least common denominator (LCD) is related to the proportion method when solving rational equations.
  • When applying the proportion method to solve a rational equation, it is often necessary to find the least common denominator (LCD) of the fractions in the equation. The LCD is the smallest positive integer that is a multiple of all the denominators, and it is used to simplify the rational expressions before setting up the proportion. Finding the LCD ensures that the denominators in the proportion are the same, which is a crucial step in the cross-multiplication process and the overall solution of the rational equation.
• Analyze how the proportion method can be advantageous compared to other techniques for solving rational equations that contain variables in both the numerator and denominator.
  • The proportion method is particularly useful when solving rational equations that have variables in both the numerator and denominator. In these cases, other methods, such as clearing denominators or using algebraic manipulation, can become increasingly complex and difficult to apply. The proportion method, on the other hand, allows you to set up a proportion between the numerators and denominators, which can simplify the equation and make it easier to solve. By identifying a common factor through the proportion, you can then solve for the variable(s) in the original rational equation more efficiently than with other techniques.

Related terms

Rational Equation:

A rational equation is an equation that contains one or more rational expressions, which are fractions with polynomial expressions in the numerator and denominator.

Cross-Multiplication:

Cross-multiplication is a method used to solve proportions by multiplying the means (the two middle terms) and setting them equal to the product of the extremes (the two outer terms).

Least Common Denominator (LCD):

The least common denominator is the smallest positive integer that is a multiple of all the denominators in a set of fractions, which is used to simplify rational expressions.