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Chapter 1: Problem 20

Solve. $$2 w+2 h+l=p, \text { for } w$$

Short Answer

Expert verified
w = \frac{p - 2h - l}{2}

Step by step solution

01

Identify the equation

The given equation is: \[2w + 2h + l = p\] The goal is to solve for \(w\).
02

Isolate the term involving w

Subtract the terms \(2h\) and \(l\) from both sides of the equation to isolate \(2w\): \[2w = p - 2h - l\]
03

Solve for w

Divide both sides of the equation by 2 to solve for \(w\): \[w = \frac{p - 2h - l}{2}\]

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Key Concepts

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

variable isolation
In this exercise, the first concept we tackle is **variable isolation**. This means focusing solely on the variable we want to solve. For example, in the given equation \(2w + 2h + l = p\), we want to isolate the variable \(w\). To do this, we aim to get \(w\) by itself on one side of the equation. To achieve variable isolation, we first identify any other terms connected to \(w\) and move them to the other side of the equation. By doing so, we eliminate unwanted terms and effectively 'isolate' \(w\). This process sets the stage for easily solving the equation. Always remember, isolating the variable is the key first step in solving linear equations.
algebraic manipulation
After deciding which variable to isolate, **algebraic manipulation** plays a crucial role. This involves using basic algebra rules to rearrange and simplify the equation. In our example, after the identification that \(2w + 2h + l = p\), we need to move \(2h\) and \(l\) to the other side to separate \(w\). So, perform simple operations like addition, subtraction, multiplication, or division. Specifically here, we subtract \(2h\) and \(l\) from both sides: \(2w = p - 2h - l\). Notice how these steps don't change the equation's balance. Each manipulation ensures the equation holds true, while also helping to isolate and solve our desired variable.
equation solving
Lastly, let's talk about **equation solving**. This connects everything we've done before, leading up to finding the value of our isolated variable. In our example with the isolated term: \(2w = p - 2h - l\). Now, we need to get \(w\) entirely by itself by performing the inverse operation. Since \(w\) is currently multiplied by 2, we divide both sides by 2 to solve for \(w\): \(w = \frac{p - 2h - l}{2}\). This gives us the solution for \(w\). Remember, solving equations involves 'undoing' operations done to the variable until it's by itself.

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Most popular questions from this chapter

Insert one pair of parentheses to convert each of the following false statements into a true statement. $$ 2 \cdot 7+3^{2} \cdot 5=104 $$

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