Problem 50 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ P=2(w+h+h) \text { for } h $$

Step-by-Step Solution

Answer

The solution is \( h = \frac{P - 2w}{4} \).

Step 1: Simplify the Formula

First, let's simplify the given equation by consolidating similar terms. Currently, the equation is: \[P = 2(w + h + h)\]Notice that \( h \) is added twice, so we can rewrite it as:\[P = 2(w + 2h)\]

Step 2: Distribute the 2 on the Right Hand Side

Next, to eliminate the parentheses, distribute the 2 across the terms inside the parentheses. This gives:\[P = 2w + 4h\]

Step 3: Isolate Terms with Variable h

We need to solve for \( h \), so we'll start by isolating terms involving \( h \). Subtract \( 2w \) from both sides of the equation:\[P - 2w = 4h\]

Step 4: Solve for h

Now, to get \( h \) by itself, divide both sides of the equation by 4:\[h = \frac{P - 2w}{4}\]

Step 5: Final Result

The equation is now solved for \( h \). The formula for \( h \) in terms of \( P \) and \( w \) is:\[h = \frac{P - 2w}{4}\]

Key Concepts

Formula Manipulation

Formula manipulation is an essential technique in algebra and mathematics overall, allowing you to reshape formulas to solve for different variables or adapt them to specific uses. This process involves strategically using mathematical operations—such as addition, subtraction, multiplication, or division—to rearrange a formula to express it differently. For instance, in the given exercise to solve the equation for the variable \( h \), you start by simplifying and then manipulating the original formula \( P = 2(w + h + h) \). Breaking it down:

• You consolidate like terms, simplifying slash condensing the equation by adding \( h + h \) to become \( 2h \).
• This is followed by distributing any constants outside parentheses, such as the 2 in this formula, across the terms inside, effectively removing the parentheses and simplifying to \( P = 2w + 4h \).

Manipulating formulas like this helps you to express relationships between variables in a more direct form, making it easier to isolate and solve for specific variables.

Isolating Variables

Isolating a variable involves rearranging an equation so that the variable of interest stands alone on one side of the equation. This method is crucial for solving equations, where you aim to find the value of a variable. In our task, the goal was to isolate \( h \) in terms of \( P \) and \( w \). To achieve this, you perform operations on the equation that allow you to group all terms containing the variable on one side. In this scenario:

• Once you reached \( P = 2w + 4h \), you needed to get only the \( h \)-dependent terms isolated.
• Subtracting \( 2w \) from both sides of the equation gives you \( P - 2w = 4h \).

Once isolated, you then solve for the variable by dividing all terms by the coefficient of that variable, like dividing everything by 4 to solve \( h = \frac{P - 2w}{4} \). Separating and solving for variables are powerful tools in understanding how elements of a formula relate and interact.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators functioning together to define a relationship or a calculation. They can appear in various forms ranging from simple equations to complex formulas. In this exercise, the expression \( P = 2(w + h + h) \) is manipulated into different equivalent forms to serve different purposes. Key components include:

• **Terms**: Each part of the expression separated by a plus or minus sign, like \( w \) and \( h \) in our findings.
• **Coefficients**: Numbers multiplied by the variables in an expression, such as the 4 in \( 4h \) in the equation \( P - 2w = 4h \).
• **Variables**: Symbols representing numbers or quantities, and in our case, \( h \) is the variable being focused on.

Understanding algebraic expressions allows you to read and decipher equations easily, ensuring that you can manipulate and solve them effectively. In our example, breaking down the expression step by step through formula manipulation and isolating techniques enables us to solve for \( h \). These skills are fundamental for anyone working with mathematical or scientific expressions.