Problem 43 - Intermediate Algebra

Question

Solve each formula for the specified variable. \(I=\frac{E}{R_{L}+r}\) for \(r\) (from physics)

Step-by-Step Solution

Answer

\( r = \frac{E - IR_L}{I} \)

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, multiply both sides of the equation by \( R_L + r \). This gives:\[ I(R_L + r) = E \]

Step 2: Distribute the Current (I)

Distribute \( I \) on the left-hand side of the equation:\[ IR_L + Ir = E \]

Step 3: Isolate the term with r

Subtract \( IR_L \) from both sides of the equation to isolate the term with \( r \):\[ Ir = E - IR_L \]

Step 4: Solve for r

Divide both sides by \( I \) to solve for \( r \):\[ r = \frac{E - IR_L}{I} \]

Key Concepts

Physics Formula

Physics formulas are essential mathematical representations used to describe physical principles and phenomena. The formula given in the exercise, \( I = \frac{E}{R_{L} + r} \), is a classic example from electricity, where \( I \) represents the current, \( E \) is the electromotive force or voltage, \( R_{L} \) is the load resistance, and \( r \) is the internal resistance. These formulas make it easier to describe the relationships between different physical quantities. By manipulating these formulas, we can find unknown values and gain deeper insights into how various physical entities interact. This foundational understanding is crucial for solving more complex physics problems.

Variable Isolation

Variable isolation is a fundamental skill in algebra that involves rearranging an equation to solve for a specific variable. In the provided exercise, we aim to isolate \( r \) in the formula \( I = \frac{E}{R_{L} + r} \). The goal is to express \( r \) alone on one side of the equation. This process generally involves the following steps:

• Breaking down the equation into simpler parts by reversing operations.
• Eliminating fractions by multiplying through by a denominator.
• Rearranging terms to group similar variables together.

By practicing variable isolation, students can enhance their problem-solving skills and apply these techniques to various mathematical and real-world problems.

Solving Equations

Solving equations involves finding the value of an unknown variable that satisfies the given mathematical expression. In this exercise, solving the equation \( I(R_{L} + r) = E \) for \( r \) involves sequentially:

• Eliminating the denominator by multiplying both sides by \( R_{L} + r \).
• Distributing \( I \) across \( R_{L} + r \) to separate the terms.
• Reorganizing the equation by moving all terms involving \( r \) to one side.
• Ultimately, isolating \( r \) by a final division operation to find its value.

Through practice, solving equations can become more intuitive, allowing students to approach complex problems with confidence and efficiency.

Intermediate Algebra

Intermediate algebra serves as a bridge between basic algebraic concepts and more advanced mathematical studies. It encompasses various concepts, such as manipulating complex formulas, solving multi-step equations, and understanding advanced functions. In the exercise, the use of distributive property and isolation of a variable are prime examples of intermediate algebra techniques in action. These skills are pivotal because:

• They help tackle real-world problems by providing a structured approach to rearranging and simplifying expressions.
• They enhance critical thinking, allowing students to make logical deductions based on algebraic manipulations.
• They prepare students for higher-level courses in mathematics and science by building a strong foundational skill set.

Mastering intermediate algebra is essential for academic progress and practical problem-solving across various disciplines.