Solve each equation, if possible. (5)/(5 z-11)+(4)/(2 z-3)=(-3)/(5-z)

Name: Problem 66, Chapter 1: Algebra and Trigonometry
Uploaded: 2021-06-13T09:17:50-08:00
Duration: 5 min 11 s
Channel: Yujie Wang
Description: Solve each equation, if possible. (5)/(5 z-11)+(4)/(2 z-3)=(-3)/(5-z)

step 1 Okay, so we want to solve this equation. First, we need to cancel out all the denominators. Okay

Solve each equation, if possible. (5)/(5 z-11)+(4)/(2...

Key Concepts

Extraneous Solutions

In the process of solving rational equations, especially after clearing denominators, it is possible to obtain extraneous solutions—answers that emerge from the algebraic manipulation but do not satisfy the original equation or fall outside of its domain restrictions. Verifying the solutions by substituting them back into the original equation is essential to ensure their validity.

Common Denominator and Clearing Fractions

A key step in solving rational equations involves finding the least common denominator (LCD) of all fractions involved. Multiplying both sides of the equation by the LCD clears the fractions, transforming the rational equation into a polynomial equation that is often easier to solve.

Domain Restrictions

When dealing with rational expressions, it is crucial to identify the domain restrictions, which are the values of the variable that would make any denominator equal to zero. These restrictions must be considered both before and after solving the equation, as potential solutions must be discarded if they are not part of the domain.

Rational Equations

This concept involves equations that contain one or more rational expressions—fractions whose numerators and/or denominators are polynomials. Solving rational equations typically requires finding a common denominator to combine terms and then clearing the fractions to form a polynomial equation that can be solved with standard algebraic methods.

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