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Chapter 2: Problem 11

Solve each formula for the specified variable. \(. A=L W\) for \(W\)

Short Answer

Expert verified
W = \frac{A}{L}

Step by step solution

01

Identify the given formula

The given formula is \(A = L \cdot W\). We need to solve this formula for the variable \(W\).
02

Isolate the desired variable

To solve for \(W\), we need to isolate \(W\) on one side of the equation. We can do this by dividing both sides of the equation by \(L\). The equation becomes \[ \frac{A}{L} = W \]
03

Simplify the equation

Rewriting the equation, we have \[ W = \frac{A}{L} \] Now, \(W\) is isolated, and the formula is solved for \(W\).

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Key Concepts

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

Formula Manipulation
Formula manipulation is the process of rearranging a mathematical formula to isolate and solve for a particular variable. This is very important in algebra because it helps you to understand different relationships between variables. For example, if you have the formula for the area of a rectangle, which is \(A = L \cdot W\), you might need to manipulate it to solve for one of its variables, like width \(W\).
This involves using basic algebraic operations, such as addition, subtraction, multiplication, and division, to rearrange the formula.
  • Identify the formula you need to work with.
  • Determine which variable you need to solve for.
  • Use algebraic steps to isolate this variable.
Manipulating formulas helps you to move terms around the equation, making it easier to isolate the desired variable.
Variable Isolation
Variable isolation means rearranging an equation so that the variable you want to solve for is on one side of the equation, and everything else is on the other. This is often the key step in solving equations. For the given formula, \(A = L \cdot W\), we are asked to solve for \(W\).
To isolate \(W\):
  • Look at the equation: \(A = L \cdot W\).
  • To get \(W\) by itself, divide both sides of the equation by \(L\).
  • This gives you: \[W = \frac{A}{L}\]
Now, \(W\) is isolated, and we have the equation \[W = \frac{A}{L}\], showing \(W\) in terms of \(A\) and \(L\). The process of isolating variables is essential in algebra because it allows us to solve for unknowns and understand the relationships between different elements in an equation.
Algebraic Equations
Algebraic equations involve variables, constants, and operational symbols (like +, -, \cdot, \div) arranged in equalities or inequalities. Solving algebraic equations is fundamental to algebra and involves finding the values of the variables that make the equation true.
In our exercise, we started with the algebraic equation \(A = L \cdot W\).
Here鈥檚 a quick recap of the steps we followed:
  • We identified that we need to solve for \(W\).
  • We divided both sides of the equation by \(L\) to isolate \(W\).
  • We simplified the equation to get \[W = \frac{A}{L}\].
This process illustrates how algebraic operations can be used to solve for a particular variable. Understanding these steps helps you to tackle more complex equations and problems. It also builds a strong foundation for higher-level math concepts.

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