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Chapter 11: Problem 83

Solve each equation for the specified variable. $$ P=2 L+2 W \text { for } W $$

Short Answer

Expert verified
W = \( \frac{P - 2L}{2} \)

Step by step solution

01

Isolate the terms with W

Start with the equation: \( P = 2L + 2W \). Subtract \( 2L \) from both sides to isolate the terms with \( W \) on one side of the equation: \( P - 2L = 2W \).
02

Solve for W

Divide both sides of the equation by 2 to solve for \( W \): \( \frac{P - 2L}{2} = W \).
03

Simplify the equation

The equation simplifies to: \( W = \frac{P - 2L}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating variables is a critical skill in algebra. It involves manipulating the equation so that the variable you are solving for stands alone on one side of the equation.
For example, in the exercise provided:
$$ P = 2L + 2W \text { for } W $$
You want to solve for the variable W. This process starts with moving other terms away from W.
In this case, subtract 2L from both sides:
$$ P - 2L = 2W $$
Now, only terms involving W are on one side of the equation. From here, you can perform further steps to isolate W completely.
Algebraic Manipulation
Algebraic manipulation techniques are the heart of solving equations. This includes adding, subtracting, multiplying, or dividing both sides of the equation. These operations help transform the equation into a simpler form.
Continuing from where we isolated the terms with W:
$$ P - 2L = 2W $$
We divide both sides by 2 to fully isolate W:
$$ \frac{P - 2L}{2} = W $$
Dividing by 2 balances the equation and simplifies it. Algebraic manipulation maintains the equation's balance, ensuring both sides remain equal.
Solving Linear Equations
A linear equation represents a relationship where each term is either a constant or the product of a constant and a single variable. Solving these equations involves using the techniques of isolating variables and algebraic manipulation.
The exercise:
$$ P = 2L + 2W \text{ for } W $$
follows this pattern. After isolating W and using algebraic manipulation, we solve it:
Step 1: Subtract 2L from both sides:
$$ P - 2L = 2W $$
Step 2: Divide by 2:
$$ \frac{P - 2L}{2} = W $$
This process reveals the value of W in simplified form:
$$ W = \frac{P - 2L}{2} $$
Mastering these methods will make solving linear equations easier and more intuitive.

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