Problem 59 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ \frac{1}{a}+\frac{1}{b}=1 \text { for } a $$

Step-by-Step Solution

Answer

The solution is \( a = \frac{b}{b-1} \).

Step 1: Clear the fractions

Multiply every term by \( ab \), (the common denominator) to eliminate the fractions:\[ab\left(\frac{1}{a}+\frac{1}{b}\right) = ab \times 1\]Simplifying the equation, we get:\[b + a = ab\]

Step 2: Rearrange the equation

Isolate the term \( a \) on one side of the equation by moving \( a \) to the right side:\[b = ab - a\]

Step 3: Factor the equation

Factor out \( a \) from the right-hand side:\[b = a(b - 1)\]

Step 4: Solve for a

Divide both sides of the equation by \( b-1 \) to find \( a \):\[a = \frac{b}{b-1}\]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the values of variables that satisfy a given equality. In our original exercise, we aimed to solve for the variable \(a\) in the equation \(\frac{1}{a}+\frac{1}{b}=1\). Here's a brief run-through of the essential steps taken in our solution:

• First, we cleared the equation of fractions by multiplying both sides by the common denominator \(ab\), which simplifies the equation considerably.
• Next, we simplified it to \(b + a = ab\).
• This allowed us to proceed with isolating the desired variable \(a\), as we will see in more detail in the following sections.

Understanding how to solve equations aids in tackling more complex mathematical problems effectively.
Breaking equations down into smaller, more manageable pieces is crucial for accurate problem solving.

Formula Transformation

Formula transformation is the process of rewriting a formula to express a variable in terms of other variables. This is particularly useful when dealing with more complex mathematical formulas. In our original exercise, transforming the formula involved:

• Rearranging the terms to isolate the target variable, \(a\).
• Moving \(a\) across the equality sign to enable factoring.

Transformations sometimes require multiple iterations, each step bringing us closer to expressing our variable of interest on one side of the equation. Performing these transformations helps us present variables in a way that is easier for specific calculations or further analysis. Additionally, mastering formula transformation enhances algebraic flexibility, allowing you to approach problems from different angles.

Isolating Variables

Isolating a variable involves rearranging an equation so that a single variable stands alone on one side of the equation. This is the key objective when solving for a specified variable in algebraic expressions. To isolate \(a\) in our problem, here’s what occurred:

• We started by obtaining a linear expression \(b = ab - a\).
• Factored out \(a\) from the right side to make \(b = a(b - 1)\).
• To complete the isolation, we divided both sides by \(b - 1\) to achieve \(a = \frac{b}{b-1}\).

Isolating variables is a critical skill as it lays the groundwork for solving equations effectively. Being skilled in isolating variables allows you to solve diverse and complex mathematical problems and is essential for higher-level algebra and calculus.