Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ x y^{2}=x-8 $$

Step-by-Step Solution

Answer

\( x = \frac{-8}{y^2 - 1} \)

Step 1: Isolate x on One Side

Given the equation \( x y^2 = x - 8 \), we start by isolating \( x \) on one side. First, subtract \( x \) from both sides to get \( x y^2 - x = -8 \).

Step 2: Factorize the Left Side

We can factor \( x \) from the left side to rewrite it as \( x(y^2 - 1) = -8 \).

Step 3: Solve for x

To solve for \( x \), divide both sides by \( y^2 - 1 \): \( x = \frac{-8}{y^2 - 1} \). This expression for \( x \) gives us the solution depending on \( y \).

Key Concepts

Factoring Expressions

Factoring is a critical tool in algebra that involves breaking down expressions into simpler components, called factors. In the given equation, we have: - Start with: \( x y^2 - x = -8 \).- Notice that both terms on the left-hand side have a common factor, which is \( x \). By factoring \( x \) out, the expression becomes: \( x(y^2 - 1) = -8 \). This is useful because it simplifies complex expressions, making it easier to solve the equation. Factoring reveals the hidden structure of the algebraic expression, enabling us to see terms that behave like simpler, more manageable entities. It often helps in solving equations more efficiently by reducing complexity early in the process.

Solving Equations

Once you understand how to factor expressions, solving the equations becomes more approachable. An equation like \( x(y^2 - 1) = -8 \) requires us to find the value of \( x \) in terms of \( y \). Solving equations typically follows this sequence:

• Isolate the variable of interest.
• Perform algebraic operations that simplify the expression.

For our equation, we want to solve for \( x \), so we divide both sides by the expression \( y^2 - 1 \). This gives us: \[x = \frac{-8}{y^2 - 1}\]This process of solving allows us to express one variable concerning others, providing an equation that can be used for further computations or substitutions in more complex problems.

Variable Isolation

Variable isolation refers to the technique of manipulating an equation to get a particular variable alone on one side of the equation. This is crucial in algebra as it helps simplify solving processes and provides explicit solutions. Let's take a closer look at its application in the problem:- Start with isolating the variable: We began with \( x y^2 = x - 8 \) and strategically subtracted \( x \) from both sides, resulting in \( x(y^2 - 1) = -8 \).- The goal is achieved by making \( x \) the subject of the formula.Isolating variables transforms a complicated expression into a straightforward solution for one variable in terms of another. This step-by-step transformation aids in the easy verification of solutions and fosters a deeper understanding of relationships among different variables. By isolating \( x \) as \( x = \frac{-8}{y^2 - 1} \), we've derived a simple formula that indicates how \( x \) changes with \( y \). This clarity often serves as the foundation for solving more advanced algebraic equations and performing further manipulations.