Problem 13 - 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? $$A=\frac{1}{2} b h \text { for } b$$

Step-by-Step Solution

Answer

The solution is \( b = \frac{2A}{h} \).

Step 1: Identify the equation

The equation given in the problem is \( A = \frac{1}{2} b h \), which represents the area of a triangle.

Step 2: Isolate \( b \) on one side of the equation

To solve the equation for \( b \), the goal is to get \( b \) by itself on one side of the equation. Start by multiplying both sides of the equation by 2 to eliminate the fraction. This results in \( 2A = b h \).

Step 3: Divide both sides by \( h \)

The final step is to get \( b \) by itself. This is accomplished by dividing both sides of the equation by \( h \). The algebraic operation yields the solution \( b = \frac{2A}{h} \).

Key Concepts

Formula Manipulation

Formula manipulation is a fundamental skill in algebra that allows us to rearrange equations to solve for a specific variable. In simple terms, it's like reshuffling the equation to isolate the term we are interested in. This reshuffling involves using the properties of equality: whatever operation you do to one side, you must do to the other.

• Identify the Equation: First, find the variable you need to solve for in the problem statement. For example, if you have the equation for the area of a triangle, \( A = \frac{1}{2} bh \), and you need to solve for \( b \).
• Operations to Isolate: Apply arithmetic operations such as addition, subtraction, multiplication, or division to both sides of the equation to isolate the variable. For instance, to clear a fraction, you might multiply by the denominator.
• Reverse Operations: If an operation is not familiar or seems complex, think of reversing it to approach the solution more readily. For example, if a term is added, consider subtracting it from both sides of the equation.

Understanding and practicing these basic algebraic manipulations can make working with formulas much easier.

Triangle Area

The area of a triangle is a basic concept in geometry that can often appear in algebra exercises. The formula to find the area of a triangle is given by \( A = \frac{1}{2} bh \), where:

• \( A \) is the area.
• \( b \) is the base of the triangle.
• \( h \) is the height of the triangle.

This formula helps calculate the surface area enclosed by the triangle's three sides. Knowing the area formula is essential for solving problems involving triangles where dimensions are not immediately clear. A visual depiction often helps, where you picture the triangle standing on its base, with the height extending from the base to the opposite vertex.
You can think of the area of a triangle as half the area of a rectangle, because that's how the formula is derived. Imagine a rectangle with the same base and height as the triangle; the triangle occupies exactly half of that space. This understanding can greatly aid in visualizing and solving problems related to triangle areas.

Variable Isolation

Variable isolation is another key skill in algebra, allowing us to solve equations by focusing on one particular variable. When you isolate a variable, you are essentially untying it from everything else in the equation.
Here's how to isolate \( b \) in the equation \( A = \frac{1}{2} bh \):

• Eliminate Fractions: Start by multiplying each side by 2. This removes the fraction, giving you \( 2A = bh \).
• Simplify: The goal is to have the variable \( b \) on its own. Divide both sides by \( h \) to penalize \( h \)'s contribution to the equation, leading to \( b = \frac{2A}{h} \).
• Check Your Work: Always review to ensure the equation makes sense. Verify that substituting your solution back into the original formula satisfies your initial condition.

The method of isolating a variable helps turn complex equations into simpler, solvable ones by systematically focusing on removing unwanted parts of the equation. With practice, this becomes a straightforward and powerful tool in algebra.