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Chapter 2: Problem 76

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

Short Answer

Expert verified
L = \( \frac{ \mathscr{A} }{ W } \).

Step by step solution

01

Understand the Given Equation

The equation to solve is \(\backslash mathscr\{A\} = L W\). The goal is to isolate the variable \(L\) on one side of the equation.
02

Isolate the Variable

To solve for \(L\), divide both sides of the equation by \(W\). This will move \(W\) to the denominator on the right side. The equation becomes: \[\frac{ \backslash mathscr\{A\} }{W} = L \]
03

Solution

Now the variable \(L\) is isolated. Therefore, \(L = \backslash frac{ \backslash mathscr\{A\} }{W}\).

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Key Concepts

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

Isolating Variables
When solving equations, one of the key steps is to isolate the variable you are solving for. This means getting the variable by itself on one side of the equation. This process typically involves undoing whatever operations are being performed on the variable.

For instance, if the variable is being multiplied by another number, you would divide to cancel out the multiplication. Similarly, if the variable is being added to something, you would subtract to isolate it.

In the example provided, we need to isolate the variable \(L\) in the equation \(\mathscr{A} = L W\). To do this, we perform the inverse operation to what's currently being done to \(L\).
Dividing Equations
Dividing both sides of an equation is a common technique used to isolate a variable. This technique will help you solve many algebraic equations, especially when a variable is being multiplied by another value.

In our equation \(\mathscr{A} = L W\), \(L\) is being multiplied by \(W\). To isolate \(L\), we need to perform the opposite operation of multiplication, which is division. We divide both sides of the equation by \(W\) to isolate \(L\).

The division gives us the following equation: \[ \frac{\mathscr{A}}{W} = L \]. By dividing both sides by \(W\), we effectively cancel out \(W\) on the right side of the equation, leaving \(L\) by itself.
Solving for a Variable
Solving for a variable involves a few straightforward steps. Once the variable is isolated, you rewrite the equation with the variable clearly defined. In our example, we started with \(\mathscr{A} = L W\) and used division to isolate \(L\).

After dividing both sides by \(W\), we got \( \frac{\mathscr{A}}{W} = L \). To finalize the solution, we can rewrite this as \( L = \frac{\mathscr{A}}{W} \).

This simple method of isolating the variable, using division, and then writing the solution clearly will help you solve most algebraic equations. Remember,
  • Identify what operation is being performed on the variable.
  • Use the inverse operation to isolate the variable.
  • Simplify the equation to find your solution.

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