Problem 45 - Intermediate Algebra

Question

Solve each formula for the specified variable. \(\mu_{R}=\frac{n_{1}\left(n_{1}+n_{2}+1\right)}{2}\) for \(n_{2}\) (from statistics)

Step-by-Step Solution

Answer

\( n_{2} = \frac{2\mu_{R} - n_{1}^2 - n_{1}}{n_{1}} \)

Step 1: Substitute the Formula

Let's start with the given formula \( \mu_{R} = \frac{n_{1}(n_{1} + n_{2} + 1)}{2} \). We want to solve for \( n_{2} \). Begin by isolating the fraction to simplify expressions later.

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by 2: \( 2\mu_{R} = n_{1}(n_{1} + n_{2} + 1) \).

Step 3: Distribute \( n_1 \)

Distribute \( n_1 \) on the right hand side: \( 2\mu_{R} = n_{1}^2 + n_{1}n_{2} + n_{1} \).

Step 4: Isolate \( n_{1}n_{2} \)

Move all other terms except for \( n_{1}n_{2} \) to the left side: \( 2\mu_{R} - n_{1}^2 - n_{1} = n_{1}n_{2} \).

Step 5: Solve for \( n_{2} \)

Divide both sides by \( n_{1} \) to solve for \( n_{2} \): \( n_{2} = \frac{2\mu_{R} - n_{1}^2 - n_{1}}{n_{1}} \).

Key Concepts

Intermediate Algebra

Intermediate algebra acts as a bridge between basic algebra and advanced mathematical topics. It features all the foundational skills needed to tackle complex equations. In intermediate algebra, you encounter concepts such as quadratic equations, factoring, and solving linear equations. These skills are crucial for manipulating expressions and isolating desired variables in equations.


You are often required to deal with expressions that involve parentheses, exponents, and fractions. Handling such expressions demands a solid understanding of distributive, associative, and commutative properties.

Understanding these principles allows you to break down complicated algebraic expressions step by step. This is essential for solving formulas or equations, like the one given in the exercise.


Note that algebra is not only about calculations. It involves logical reasoning and the ability to transform equations into simpler forms. This logical flow goes hand in hand with problem-solving skills that algebra helps to develop.

Statistics

Statistics is a branch of mathematics focused on the collection, analysis, interpretation, and presentation of numerical data. It's used to make informed decisions or predictions in various fields, such as science, economics, and social studies.


In the context of the provided exercise, statistical formulas are key in understanding and estimating parameters such as means, variances, and rates. These parameters summarize data sets and allow for informed conclusions based on sample data.

The expression given in the problem involves statistical notation where parameters like \( \mu_{R} \) (a mean or average) play a significant role.

Statistical equations are particularly important when working with probabilities and predictions. They help to determine relationships and trends found in data, aiding in the analysis and understanding of the world through numbers.

Understanding statistical concepts and how to translate real-world problems into statistical equations is fundamental in deriving insightful inferences from data.

Variable Isolation

Isolating a variable is a core technique in algebra and broader mathematical problem-solving processes. It involves manipulating an equation to express one specific variable distinctly on one side of the equation, making it straightforward to identify its value.


In the provided exercise, the objective was to isolate \( n_{2} \). This process is achieved by performing inverse operations and strategically rearranging the terms.

Here’s a quick breakdown of the isolation procedure:

• Start by eliminating fractions to reduce complexity. This involves multiplying both sides of the equation by the denominator.
• Next, distribute any coefficients across terms to simplify the relationships between the variables.
• Rearrange the equation to move all terms except the one you are solving for across to one side of the equation.
• Finally, divide or multiply to isolate the desired variable.

Through practice, variable isolation becomes second nature. It plays a fundamental role not only in algebra but also in solving any mathematics-based real-world problems.