Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=\frac{1}{2} h(a+b) \text { for } a$$

Step-by-Step Solution

Answer

The formula solved for \(a\) is \(a = \frac{2A - bh}{h}\).

Step 1: Multiply both sides by 2

Multiplying the whole equation by 2 negates the fraction on the right side of the equation, resulting in \(2A = h(a+b)\).

Step 2: Distribute \(h\)

Now distribute \(h\) to both \(a\) and \(b\), which results in \(2A = ah + bh\).

Step 3: Isolate \(a\)

Subtract \(bh\) from both sides of the equation to isolate \(a\). The resulting formula is \(a = \frac{2A - bh}{h}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging terms in an equation or expression to make it simpler or to solve for a particular variable. It's a game of balance, ensuring that whatever operation is performed on one side of the equation is also applied to the other side to maintain equality.

For instance, in the provided exercise where the area of a trapezoid is represented as \(A = \frac{1}{2} h(a+b)\), algebraic manipulation is used to solve for the variable \(a\). The initial step multiplies both sides of the equation by 2. This is a strategic move to eliminate the fraction, which can complicate further steps. Here's how it's done in LaTeX: Multiply the entire equation by 2 to obtain \(2A = h(a+b)\).

Subsequent steps then involve distributing terms and isolating the desired variable, all of which fall under the umbrella of algebraic manipulation. Understanding how to wield these manipulation techniques is crucial, as they are the basic tools for solving not just simple equations but complex mathematical problems as well.

Isolating Variables

Isolating variables in an equation is a fundamental skill in algebra that involves getting a variable alone on one side of an equation. The purpose of isolating a variable is to find its value or to express it in terms of other variables.

In our exercise, the goal is to solve for \(a\), which means clearing it from other terms. After multiplying both sides by 2 and distributing \(h\), the equation resembles \(2A = ah + bh\). To isolate \(a\), we can remove \(bh\) from both sides. This is achieved by subtracting \(bh\) from each side, demonstrating a critical concept: whatever you do to one side, you must do to the other.

The equation transforms into \(2A - bh = ah\). To completely isolate \(a\), divide both sides by \(h\), resulting in \(a = \frac{2A - bh}{h}\). This isolating step is necessary to understand the relationship between \(a\) and the other variables - \(A\) and \(b\), in the context of a given formula like the trapezoid area formula.

Mathematical Equations

Mathematical equations are statements of equality that showcase the relationship between different mathematical expressions. They act as the backbone of algebra and are found in nearly all aspects of more advanced mathematics.

Our example uses the equation of the area of a trapezoid, a classic geometric formula. The original equation \(A = \frac{1}{2} h(a+b)\) speaks to the relationship between the trapezoid's area (\(A\)), height (\(h\)), and the sum of the lengths of its bases (\(a\) and \(b\)). Through algebraic manipulation and variable isolation, we rewrote the equation to express \(a\) in terms of \(A\), \(b\), and \(h\), providing insight into how these variables are interconnected.

Problem-solving with mathematical equations often involves identifying what is known and what is unknown, then using manipulation techniques to find the unknowns. The clarity gained through solving an equation can deepen one's understanding of the concepts at hand, like recognizing a formula and understanding its application in practical situations.