Problem 64 - Precalculus

Question

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

Step-by-Step Solution

Answer

The solution for \(x\) is \(\frac{-a^3 - b^3}{ab - a^2 - b^2}\).

Step 1: Expand the Terms

Distribute the fractions inside the parentheses. Rewrite the expression:\[ 1 - \frac{a}{b} + \frac{a^2}{bx} - \frac{b}{a} + \frac{b^2}{ax} = 0 \]

Step 2: Combine Like Terms

Combine all the fractions over a common denominator of \(bx\):\[ \frac{abx - a^2x + a^3 - b^2x + b^3}{abx} = 0 \]

Step 3: Multiply to Clear Denominator

Multiply through by \(abx\) to clear the fraction:\[ abx - a^2x + a^3 - b^2x + b^3 = 0 \]

Step 4: Rearrange Terms

Arrange the terms such that all terms involving \(x\) are on one side:\[ abx - a^2x - b^2x = -a^3 - b^3 \]

Step 5: Factor Out the Common Variable

Factor out \(x\) from the left-hand side:\[ x(ab - a^2 - b^2) = -a^3 - b^3 \]

Step 6: Solve for x

Divide both sides by \((ab - a^2 - b^2)\) to solve for \(x\):\[ x = \frac{-a^3 - b^3}{ab - a^2 - b^2} \]

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. In this exercise, the primary objective is to express one variable, such as \(x\), in terms of the other variables involved. This often requires:

• Expanding expressions
• Rewriting terms
• Combining like terms
• Performing basic arithmetic operations

Through careful algebraic manipulation, you simplify the equation step by step until you isolate the desired variable. It is crucial to perform these manipulations correctly, as making mistakes here can lead to incorrect solutions.

Common Denominators

Finding a common denominator is essential when dealing with equations that include fractions. It allows us to combine different fractional expressions into a single fraction. In this equation-solving process, after distributing the terms inside the parentheses, fractions appeared:

• \(\frac{a^2}{bx}\)
• \(\frac{b^2}{ax}\)

The common denominator for these fractions is \(abx\), which is derived from multiplying the denominators \(bx\) and \(ax\). Having a common denominator allows us to add or subtract these fractions, simplifying the equation into a singular form that can be further manipulated.

Factoring

Factoring is a technique to express an equation in a simplified and more manageable form by pulling out common elements. In step 5 of the solution, after arranging the terms on one side, factoring becomes necessary:The expression \(abx - a^2x - b^2x\) can have \(x\) factored out because it is a common factor in all terms. This results in:

• \(x(ab - a^2 - b^2)\)

By factoring, you reveal the underlying structure of the equation, which makes it easier to further solve for your variable of interest, as we will see in the next steps.

Solving for a Variable

Solving for a specific variable, such as \(x\), involves isolating it on one side of the equation. In this exercise, this is achieved after factoring by dividing both sides of the equation by the term \((ab - a^2 - b^2)\) that accompanies \(x\).Initially, after factoring we have:

• \(x(ab - a^2 - b^2) = -a^3 - b^3\)

To isolate \(x\), you divide both sides by the entire expression adjacent to \(x\):

• \(x = \frac{-a^3 - b^3}{ab - a^2 - b^2}\)

This final step confirms the solution, offering the clear expression of \(x\) in terms of the other variables present in the equation. Be meticulous during this step to ensure accuracy.