Problem 71 - Elementary Algebra

Question

Solve for the indicated variable. Solve for \(w: P=2(l+w)\).

Step-by-Step Solution

Answer

w = (P - 2l) / 2

Step 1: Distribute the 2

Start with the equation given: \[ P = 2(l + w) \]Distribute the 2 across the terms inside the parentheses:\[ P = 2l + 2w \]

Step 2: Isolate terms with the variable w

We need to isolate \(2w\). Begin by subtracting \(2l\) from both sides:\[ P - 2l = 2w \]

Step 3: Solve for w

Divide everything by 2 to solve for \(w\):\[ w = \frac{P - 2l}{2} \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. Let’s break down what it means to solve equations. In general, an equation is like a balance scale. Each side of the equation must balance out. If you do something to one side, you must do the exact same thing to the other side.
To solve for a variable, you aim to isolate it on one side of the equation. This often involves performing a series of inverse operations. For example, you might need to add or subtract terms from both sides, or multiply or divide terms. In our exercise, the goal was to find the value of \( w \) in the equation \( P = 2(l + w) \). Solving it step-by-step helps to make complex problems more manageable.

Distributive Property

The distributive property is a useful algebraic property that allows you to manipulate equations for easier solving. Specifically, it is used when you need to eliminate parentheses. The basic idea is to multiply each term inside a set of parentheses by a factor outside of the parentheses. This is best illustrated by the formula: \( a(b + c) = ab + ac \).
In the exercise, the distributive property was applied to \( 2(l + w) \). Here's how it works: instead of calculating \( l + w \) first and then multiplying by 2, you multiply each term separately, getting \( 2l + 2w \). This step is crucial because it simplifies the equation, making it easier to further solve for the variable \( w \). Always remember, the distributive property is about spreading the multiplication over addition or subtraction inside the parentheses.

Isolation of Variables

Isolation of variables is the step in solving equations where you "free" the variable you are solving for, from all other terms. This means getting the variable all by itself on one side of the equation, so you can determine its value. The key here is to perform operations that move or eliminate other terms.
In this particular task, we wanted to isolate \( w \). After using the distributive property, we had the equation \( P = 2l + 2w \). To isolate \( w \), you subtract \( 2l \) from both sides of the equation, resulting in \( P - 2l = 2w \). Then, to get \( w \) by itself, divide everything by 2, giving the final result: \( w = \frac{P - 2l}{2} \).

• Start by removing terms added or subtracted from the variable.
• Use multiplication or division to eliminate coefficients attached to the variable.

This systematic approach is essential in algebra to solve for unknowns efficiently, especially in equations with multiple steps.