Problem 3 - Prealgebra and Introductory Algebra

Question

In solving \(I=\frac{E}{R+r}\) for \(R,\) the goal is to get alone on one side of the equation.

Step-by-Step Solution

Answer

The equation \(I=\frac{E}{R+r}\), when solved for \(R\), gives \(R=\frac{E-Ir}{I}\).

Step 1: Reverse the Equation

Instead of considering it as \(I=\frac{E}{R+r}\), think of it as \(\frac{E}{R+r}=I\). This doesn't change the equation, just gives a different perspective.

Step 2: Cross-multiply

Cross-multiplication is necessary because we want to get rid of the fraction. By multiplying both sides by \(R+r\), the equation becomes \(E=I(R+r)\).

Step 3: Distribute I

Distribute the \(I\) on the right-hand side of the equation to both \(R\) and \(r\), resulting in \(E=IR+Ir\). This step is done to separate \(R\) in the equation.

Step 4: Isolate R

To isolate \(R\), first subtract \(Ir\) from both sides which gives \(E-Ir=IR\). Finally, divide both sides by \(I\) to make \(R\) the subject, then you end up with \(R=\frac{E-Ir}{I}\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to eliminate fractions in an equation. It helps simplify equations by getting rid of denominators. This can make it easier to isolate variables.
In our example, the equation starts as \( I = \frac{E}{R+r} \). To get rid of the fraction, we use cross-multiplication, which involves multiplying both sides of the equation by the denominator \((R+r)\), resulting in \(E = I(R+r)\).
Here are a few steps to keep in mind when using cross-multiplication:

• Identify the fraction in the equation you need to eliminate.
• Cross-multiply by multiplying each side of the equation by the denominator.
• Simplify the resulting equation to remove all fractions.

By eliminating the fraction, you have a linear equation in which other techniques, such as distribution and isolating variables, can be used to find the required variable.

Isolating Variables

Isolating variables is a fundamental technique in algebra. The main goal is to get the variable you are solving for alone on one side of the equation. This involves using algebraic operations to manipulate the equation.
Starting with our cross-multiplied equation \(E = IR + Ir\), our objective is to solve for \(R\).
First, subtract \(Ir\) from both sides to get \(E - Ir = IR\).
Then, you divide everything by \(I\) to finally isolate \(R\) and find \(R = \frac{E - Ir}{I}\).
Here is a step-by-step walkthrough:

• Determine which variable you want to isolate.
• Use addition or subtraction to move all other terms to the opposite side of the equation.
• Apply division or multiplication to both sides to get the desired variable alone.
• Check your work to ensure the variable is truly isolated and the operations performed are correct.

Every equation might require unique steps based on its structure, but the objective always remains the same: to isolate the variable.

Distributive Property

The distributive property is a core concept in algebra that allows you to multiply a single term by each term inside a parenthesis. It is often employed to simplify expressions and equations. According to the distributive property, \(a(b + c) = ab + ac\).
In our exercise, after cross-multiplication, the equation was \(E = I(R + r)\). To simplify this, distribute the \(I\) across the terms inside the parenthesis, yielding \(E = IR + Ir\).
Key steps when using the distributive property:

• Identify expressions that involve multiplication distributed over addition or subtraction.
• Multiply each term within the parenthesis by the term outside the parenthesis.
• Ensure each term inside the parenthesis is accounted for and multiplied properly.
• Simplify the resulting expression if possible.

Applying the distributive property helps express equations in a linear form, which makes steps like isolating variables much more straightforward.