Problem 47 - Elementary Algebra

Question

For Problems 43-54, solve each formula for the indicated variable. (Before doing these problems, cover the right-hand column and see how many of these formulas you recognize!) (Objective 2) $$ P=2 l+2 w \text { for } w $$

Step-by-Step Solution

Answer

\( w = \frac{P - 2l}{2} \)

Step 1: Identify the given formula and the variable to solve for

The given formula is \( P = 2l + 2w \) and we need to solve for the variable \( w \). This means we want to isolate \( w \) on one side of the equation.

Step 2: Rearrange the formula

To isolate \( w \), we first need to remove the term involving \( l \) from the right-hand side. Subtract \( 2l \) from both sides of the equation: \[ P - 2l = 2w \]

Step 3: Solve for w

Now that we have \( 2w \) on one side, divide every term by 2 to solve for \( w \): \[ w = \frac{P - 2l}{2} \]

Key Concepts

Solving for Variables

Solving for a specific variable in an equation means making the variable of interest stand alone on one side of the equation. It is a fundamental skill needed in algebra, ensuring you can rewrite equations in terms of different variables. Let's consider the example: you have the equation \( P = 2l + 2w \) and you need to solve for \( w \). This requires transforming the original equation to express \( w \) explicitly by itself. The goal is to make \( w \) appear by itself on one side of the equation, often the left side, while all other terms remain on the opposite side. This process can involve several steps, including addition, subtraction, multiplication, or division. Through careful application of these operations, you can isolate \( w \), uncovering its dependence on the other terms.

Isolation of Variables

When we talk about the isolation of variables, we mean positioning the variable all by itself on one side of the equation. In the context of our equation \( P = 2l + 2w \), isolating \( w \) involves two primary steps.

• Removing Extra Terms: Initially, we must get rid of any terms that are not part of the variable we want to isolate. Here, it implies subtracting \( 2l \) from both sides, turning the equation into \( P - 2l = 2w \). This step neutralizes the \( l \)-related term, effectively simplifying the equation.
  
• Adjusting the Coefficient: If the variable is multiplied by a number, as \( w \) is by 2, divide every term by this number. Thus, dividing every term by 2 gives \( w = \frac{P - 2l}{2} \), neatly isolating the variable.
  

These operations systematically enable one side of the equation to perfectly represent \( w \) as a function of the other variables.

Formula Manipulation

Manipulating formulas is a crucial algebraic skill that allows you to rearrange equations to suit different purposes. This could involve solving for unknowns or expressing a quantity in terms of other variables.
In our exercise, manipulating the formula \( P = 2l + 2w \) to solve for \( w \) demonstrates this process. Each manipulation obeys the fundamental principle that whatever operation you perform on one side, must also be performed on the other side to maintain equality.

• Start by recognizing what needs to be isolated and what can be adjusted to achieve this.
  
• Use inverse operations to counteract existing operations, such as subtraction to remove an added term.
  
• Don't forget to simplify where possible, dividing terms to solve for variables succinctly,

These methods can be applied to diverse equations, facilitating better understanding of the relationships between variables.