Problem 72 - College Algebra Essentials

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=2 k w+ \quad 2 l h+ \quad 2 w h \text { for } h$$

Step-by-Step Solution

Answer

\( h = \frac{(A - 2 k w)}{(2l + 2w)} \)

Step 1: Identify the terms containing \( h \)

First, locate all terms in the equation that contain \( h \). In this equation, two terms involve \( h \): \( 2 l h \) and \( 2 w h \)

Step 2: Move other terms to the other side of the equation

Subtract \( 2 k w \) from both sides of the equation to get: \( A - 2 k w = 2 l h + 2 w h \)

Step 3: Factor out \( h \)

Factor out \( h \) on the right side of the equation, which gives: \( A - 2 k w = h (2l + 2w) \)

Step 4: Isolate \( h \)

Finally, divide both sides of the equation by \( 2l + 2w \) to isolate \( h \): \( h = \frac{(A - 2 k w)}{(2l + 2w)} \)

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a foundational skill in mathematics, particularly in algebra. It involves rearranging, simplifying, or rewriting algebraic expressions using a variety of operations and properties. Mastery of algebraic manipulation allows students to solve equations, simplify expressions, and understand how changes to one part of an equation can affect its overall value.

When solving an equation for a variable, it's important to perform operations that will simplify the equation without changing its equality. This includes combining like terms, distributing multiplication across parentheses, adding or subtracting terms on both sides, and using the properties of equality to keep the equation balanced. In our example, subtracting the term containing another variable, which is not part of the terms we aim to isolate, simplifies the expression and prepares it for further steps in the solution process.

Factoring Expressions

Understanding how to factor expressions is crucial when working with algebraic equations. Factoring involves breaking down a complex expression into simpler parts that, when multiplied together, give back the original expression. It's akin to finding what ingredients were mixed together to create a particular dish.

In solving our exercise, factoring is used after moving terms not containing our variable of interest to the other side. We see that both remaining terms have a common variable, which in this case is h. By factoring out h, we create a clearer path to isolate the variable. This simplification effectively reduces the complexity of the equation and moves us closer to the solution.

Isolate Variable

To isolate a variable means to get the variable by itself on one side of the equation, with all other terms on the opposite side. The goal is to express the variable as a function of the other variables or constants in the equation. This allows us to clearly understand the relationship and dependency between the variables.

Continuing from the previous steps, to isolate h we divide both sides of the equation by the factored expression, which is (2l + 2w). This leaves h on one side, and expression involving only known quantities or other variables on the other side. Through this process, we achieve our objective: the variable of interest, h, is isolated and can now be expressed explicitly in terms of the other variables in the equation, demonstrating the core concept in practice.