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To eliminate the denominators and solve multiply both sides of the equation by Paragraph BI U A/ مداع T 2x = x+1 + 3 2 3x + 2 you would ... K

To eliminate the denominators and solve multiply both sides of the equation by Paragraph BI U A/ مداع T 2x = x+1 + 3 2 3x + 2 you would ... K

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Solving Rational Equations by Eliminating Denominators To eliminate the denominators and solve the equation \[ \frac{x}{5} - \frac{2x}{3} = \frac{x + 1}{2} + \frac{3x}{6}, \] you would multiply both sides of the equation by _____. In solving equations that involve fractions, the typical technique to simplify the process involves eliminating the denominators by finding and using the least common denominator (LCD). This approach transforms the equation into a polynomial equation, which is generally easier to solve. Here, the denominators are 5, 3, 2, and 6. 1. Identify the least common denominator (LCD) of these fractions. The LCD is the smallest number that each of the denominators divides without a remainder. 2. Multiply every term in the equation by the LCD. This step will eliminate the fractions by normalizing all denominators to 1. 3. After eliminating the denominators, simplify and solve the resulting equation. **To find the LCD:** - Write the prime factorization of each denominator: - 5 is a prime number (5) - 3 is a prime number (3) - 2 is a prime number (2) - 6 = 2 × 3 - The LCD is the product of the highest powers of all prime numbers appearing in the factorizations: \( LCD = 2^1 \times 3^1 \times 5^1 = 30 \) So, the least common denominator (LCD) of 5, 3, 2, and 6 is 30. ### Conclusion You would multiply both sides of the equation by **30** to eliminate the denominators and simplify the equation for solving.
Transcribed Image Text:### Solving Rational Equations by Eliminating Denominators To eliminate the denominators and solve the equation \[ \frac{x}{5} - \frac{2x}{3} = \frac{x + 1}{2} + \frac{3x}{6}, \] you would multiply both sides of the equation by _____. In solving equations that involve fractions, the typical technique to simplify the process involves eliminating the denominators by finding and using the least common denominator (LCD). This approach transforms the equation into a polynomial equation, which is generally easier to solve. Here, the denominators are 5, 3, 2, and 6. 1. Identify the least common denominator (LCD) of these fractions. The LCD is the smallest number that each of the denominators divides without a remainder. 2. Multiply every term in the equation by the LCD. This step will eliminate the fractions by normalizing all denominators to 1. 3. After eliminating the denominators, simplify and solve the resulting equation. **To find the LCD:** - Write the prime factorization of each denominator: - 5 is a prime number (5) - 3 is a prime number (3) - 2 is a prime number (2) - 6 = 2 × 3 - The LCD is the product of the highest powers of all prime numbers appearing in the factorizations: \( LCD = 2^1 \times 3^1 \times 5^1 = 30 \) So, the least common denominator (LCD) of 5, 3, 2, and 6 is 30. ### Conclusion You would multiply both sides of the equation by **30** to eliminate the denominators and simplify the equation for solving.
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