Question

Solve each equation. $$ \frac{2}{y}+\frac{5}{2}=4-\frac{2}{3 y} $$

   Solve each equation.
$$
\frac{2}{y}+\frac{5}{2}=4-\frac{2}{3 y}
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 1, Problem 23

Instant Answer

Solved by Expert Erika Bustos

07/17/2021

Step 1

This gives us: $$ 6y \cdot \frac{2}{y} + 6y \cdot \frac{5}{2} = 6y \cdot 4 - 6y \cdot \frac{2}{3y} $$ Show more…

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Solve each equation. $$ \frac{2}{y}+\frac{5}{2}=4-\frac{2}{3 y} $$

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Key Concepts

Rational Equations

Rational equations are equations that involve ratios of polynomials, where the variable can appear in the denominators. These equations often require additional care, such as determining the valid domain of solutions since any value that makes a denominator zero must be excluded from the solution set.

Clearing Fractions

Clearing fractions involves eliminating the denominators of an equation by multiplying both sides by the least common denominator (LCD). This process converts a rational equation into a polynomial equation, which simplifies the process of isolating and solving for the variable.

Domain Restrictions

Domain restrictions are an essential consideration when solving rational equations because any potential solution that results in a zero denominator must be discarded. Identifying these restrictions early on prevents arriving at extraneous solutions that do not satisfy the original equation.

Algebraic Manipulation

Algebraic manipulation involves applying various operations such as addition, subtraction, multiplication, division, and factoring to both sides of an equation. In the context of rational equations, it is crucial for rearranging the equation and isolating the variable after clearing fractions, ensuring that the steps lead to a correct and simplified solution.

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