Problem 79 - Intermediate Algebra

Question

Solve for the specified variable. $$ E=I R+I r \quad \text { for } R $$

Step-by-Step Solution

Answer

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

Step 1: Identify the equation

The equation given is \( E = IR + Ir \) and we need to solve for the variable \( R \).

Step 2: Factor out the common term

Both terms on the right side of the equation contain the variable \( I \), so we can factor \( I \) out: \( E = I(R + r) \).

Step 3: Isolate the specified variable

To solve for \( R \), divide both sides of the equation by \( I \): \( \frac{E}{I} = R + r \).

Step 4: Solve for \( R \)

Subtract \( r \) from both sides to isolate \( R \): \( R = \frac{E}{I} - r \).

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that simplifies equations by revealing common elements. If you look at an expression and notice that several terms share a common factor, you can "factor out" this commonality. This effectively reduces the complexity of the expression and makes further manipulation easier. In our exercise, the equation \( E = IR + Ir \) can be simplified by factoring out \( I \), since it appears in both terms on the right-hand side. This gives us the expression \( I(R + r) \). By doing this, you break down the equation into a simpler form without changing its value.

This prepares the equation for the next steps of solving for the variable and helps clarify the structure of the problem. When you see similar structures in algebra, always check for common factors as a primary step.

Algebraic Manipulation

Algebraic manipulation involves using a variety of strategies to rearrange and simplify equations, making it easier to isolate a specific variable. Once we've factored out the common term, in our case \( I \), the equation becomes \( E = I(R + r) \).

At this point, our aim is to get the equation in a form where we can easily solve for \( R \). First, we divide both sides by \( I \) resulting in \( \frac{E}{I} = R + r \). This step is crucial as it isolates the group \( R + r \) from the factor \( I \).

• Dividing by \( I \) helps in breaking the equation down further.
• It requires the understanding that when we multiply or divide an equation by the same non-zero value, the equation's balance remains unchanged.

Mastering algebraic manipulation is critical as it forms the backbone of solving complex algebraic equations.

Isolation of Variables

The goal of isolating variables is to have \( R \) on one side of the equation for clarity and simplicity of solution. Once the equation is simplified to \( \frac{E}{I} = R + r \), you further simplify by ensuring \( R \) stands alone.

To do this, subtract \( r \) from both sides of the equation, which transforms it into \( R = \frac{E}{I} - r \). This operation ensures that \( R \) is isolated, and the equation provides a clear solution showing \( R \) in terms of \( E \), \( I \), and \( r \).

Key elements to remember about the isolation of variables include:

• Moving terms across the equation relies on the principle of doing the same operation on both sides.
• Aim to reduce the equation step by step.

Isolating variables is a critical skill, especially in complex equations, making solutions more intuitive and easier to interpret.