Problem 47 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ h=\frac{2 A}{b+d} \text { for } A $$

Step-by-Step Solution

Answer

The formula for \(A\) is \(A = \frac{h(b + d)}{2}\).

Step 1: Recognize the Formula

We are given the formula \(h=\frac{2A}{b+d}\), which relates the variables \(h\), \(A\), \(b\), and \(d\). The task is to solve this formula for the variable \(A\).

Step 2: Eliminate the Denominator

To isolate \(A\), first multiply both sides of the equation by \(b + d\) to eliminate the denominator. This gives:\[h(b + d) = 2A\]

Step 3: Isolate the Variable

Divide both sides of the equation by 2 to solve for \(A\):\[A = \frac{h(b + d)}{2}\]

Key Concepts

Formula Manipulation

Formula manipulation is often the first step in solving algebraic problems. It involves rearranging an equation or expression to suit our needs, typically to isolate a particular variable. In the given exercise, the formula is manipulated to solve for the variable \(A\). Let's break down the manipulation process:

• Identify the structure of the formula. Here, we have the expression \(h = \frac{2A}{b+d}\).
• Decide on a strategy to isolate the desired variable, which requires a proper understanding of the interplay between the variables and arithmetical operations involved.
• Proceed by applying mathematical operations systematically—multiplying, dividing, adding, or subtracting terms to simplify and isolate the variable of interest.

Formula manipulation often serves as a bridge, allowing us to transform a complex equation into a familiar form that is easier to work with. It requires an understanding of arithmetic operations and the ability to apply them appropriately.

Solving Equations

Solving equations is a fundamental task in algebra that involves finding the value of an unknown variable that makes the equation true. The process requires logical reasoning and a strong grasp of algebraic principles.In this exercise, we start with the equation:\[h = \frac{2A}{b+d}\]Our goal is to find \(A\) in terms of \(h, b,\) and \(d\). Here's the solving process for the equation:

• Clear any fractions: Multiply through by the common denominator \(b+d\), effectively removing the fraction and simplifying the equation to \(h(b+d) = 2A\).
• Rearrange the equation: This step may involve distributing any terms as necessary to simplify further if needed—in our case, it isn't necessary until after variable isolation.

By understanding the underlying logic of each step as you solve equations, you can solve not only familiar forms but also more complex and unfamiliar ones.

Variable Isolation

Variable isolation is about re-arranging an equation such that the desired variable is on one side by itself. This is essential when you need to express one variable in terms of others.In the context of the exercise, isolating \(A\) requires these steps:

• Identify the operations affecting the target variable: In \(h(b+d) = 2A\), notice that \(2A\) suggests a multiplication by 2 on \(A\).
• Clear out other variables or constants: Divide both sides of the equation by 2 to solve for \(A\), resulting in \(A = \frac{h(b+d)}{2}\).

The technique of isolating a variable is useful in a wide variety of mathematical problems, providing straightforward solutions by returning focus to the target variable's dependency on other terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a particular value or relationship. Understanding these expressions is crucial for effectively manipulating and solving formulas.In the problem,\(h = \frac{2A}{b+d}\), is an algebraic equation with divisions and multiplications involving variables and constants.

• Terms: The expression contains several terms such as \(2A\) and \(b + d\), each representing a distinct component of the equation.
• Operations: Operations like multiplication and division connect these terms, demonstrating relationships among the variables.

Algebraic expressions form the basis of formulating equations. Being able to decipher expressions allows you to manipulate them better, paving the way for effective problem-solving.