Problem 60 - Precalculus

Question

Solve each equation for \(x\) in terms of the other letters. $$\frac{b}{a x-1}-\frac{a}{b x-1}=0, \text { where } a \neq b$$

Step-by-Step Solution

Answer

\(x = \frac{b - a}{b^2 - a^2}\)

Step 1: Identify the Common Denominator

The given equation is \( \frac{b}{a x - 1} - \frac{a}{b x - 1} = 0 \). The common denominator is \((a x - 1)(b x - 1)\). Multiply each term by this common denominator to eliminate the fractions.

Step 2: Eliminate the Fractions

After multiplying by the common denominator \((a x - 1)(b x - 1)\), we have:\[ b(b x - 1) - a(a x - 1) = 0 \] which simplifies to \[ b^2 x - b - a^2 x + a = 0 \].

Step 3: Combine Like Terms

Combine the terms involving \(x\):\[ (b^2 - a^2)x = b - a \].

Step 4: Solve for x

To solve for \(x\), divide both sides by \((b^2 - a^2)\):\[ x = \frac{b - a}{b^2 - a^2} \].

Key Concepts

Solving Equations

When solving algebraic equations, the main goal is to find the value of the unknown variable, in this case, \(x\), that satisfies the equation. The given problem is \(\frac{b}{a x - 1} - \frac{a}{b x - 1} = 0\). Start by identifying what makes the equation equal to zero. Our main focus is to manipulate the equation to isolate \(x\) on one side. By doing so, we'll express \(x\) in terms of other variables involved. Let's break down the process step by step.

Common Denominator

In equations involving fractions, it's crucial to find a common denominator to simplify the equation. For the given equation, we have the fractions \(\frac{b}{a x - 1}\) and \(\frac{a}{b x - 1}\). To eliminate the fractions and simplify the solving process, identify the common denominator which is \((a x - 1)(b x - 1)\). By multiplying every term in the equation by this denominator, we ensure the denominators of the fractions cancel out, leaving us with an equation without fractions.

Fraction Elimination

After establishing the common denominator, the next step is to eliminate the fractions by multiplication. When we multiply both terms by the common denominator \((a x - 1)(b x - 1)\), the fractions disappear. This simplifies the equation to \(b(b x - 1) - a(a x - 1) = 0\). Without fractions, the equation becomes easier to handle. When rewritten and simplified, it transforms to \(b^2 x - b - a^2 x + a = 0\), a simpler form that sets the stage for the next steps.

Combining Like Terms

The next step involves combining like terms to further simplify the equation. In \(b^2 x - b - a^2 x + a = 0\), notice the terms containing \(x\). Combine these terms: \((b^2 - a^2)x\), which collates the \(x\) terms, bringing us closer to isolating \(x\). The equation now is \((b^2 - a^2)x = b - a\). By organizing the equation in this manner, it makes it clear how to solve for \(x\) by isolating it on one side of the equation. The final expression for \(x\) is obtained by dividing both sides of the equation by \((b^2 - a^2)\), resulting in \(x = \frac{b - a}{b^2 - a^2}\).