Problem 24 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(S=4 l w+2 w h\) for \(h\)

Step-by-Step Solution

Answer

h = \frac{S - 4lw}{2w}

Step 1: Isolate Terms with h

Start with the equation \( S = 4lw + 2wh \). We want to solve for \( h \), so we need to isolate the term containing \( h \). Subtract \( 4lw \) from both sides to get \( S - 4lw = 2wh \).

Step 2: Solve for h

Now that we have \( 2wh = S - 4lw \), divide both sides by \( 2w \) to solve for \( h \). The equation becomes \( h = \frac{S - 4lw}{2w} \).

Key Concepts

Formulas in Algebra

Formulas are mathematical expressions that describe relationships between different variables. They form the backbone of algebra by allowing us to express complex ideas with simple, concise expressions.
For example, the formula given in the exercise, \(S = 4lw + 2wh\), establishes a relationship between the variables \(S\), \(l\), \(w\), and \(h\). Each of these symbols represents a particular value or measurement.

• \(S\) might stand for a surface area.
• \(l\) could represent length.
• \(w\) may indicate width.
• \(h\) possibly describes a height.

Understanding how formulas work allows us to manipulate and solve them to find unknown values. It’s important to recognize that formulas are like rules that establish how different variables are interconnected. If you wanted to change anything about one variable, you would see how it affects the others according to the formula provided.

Solving Equations

Solving equations is one of the key skills in algebra. It involves finding the values of variables that make an equation true. In the provided exercise, we were tasked with solving the formula for \(h\).
To tackle this, you need a methodical approach typically involving these steps:

• Identify the terms that contain the variable you want to solve for (in this case, \(h\)).
• Isolate this variable by performing appropriate operations (adding, subtracting, multiplying, or dividing) on both sides of the equation.
• Once isolated, solve by simplifying the expression.

In our specific problem, the term with \(h\) was \(2wh\). We subtracted \(4lw\) from both sides to start isolating \(h\).
After getting \(2wh = S - 4lw\), we divided by \(2w\) to finally solve for \(h\). This systematic approach ensures you can consistently find the values of variables in any equation reliably.

Understanding Variables

Variables are symbols used to represent numbers in mathematical expressions and equations. They act as placeholders for values that we either do not know yet or can change. This makes them incredibly powerful for generalizing problems in algebra.
In our exercise, we had variables like \(l\), \(w\), and \(h\). These symbolize unknowns or measurements that could vary depending on specific conditions.

• Variables allow us to work with general formulas instead of specific numbers, making equations flexible and applicable to numerous situations.
• In algebra, operations are performed on these variables and constants to find solutions or create relationships.
• Understanding how to manipulate these variables is crucial for solving equations and deriving solutions.

When working with variables, it’s important to remember they can represent different things in different contexts, so always pay attention to how they are defined in your particular problem or equation. This foundational understanding helps build more complex algebraic reasoning later.