91Ó°ÊÓ

Algebraic manipulation is a key skill in solving formulas. It involves rearranging and simplifying expressions to make them easier to understand or solve. Here’s how it applies to our problem:

The given formula is: \(h = \frac{2 \, \mathscr{A}}{B + b}\).
To solve for \(\mathscr{A}\), we need to perform algebraic manipulations to isolate \(\mathscr{A}\) on one side. This usually involves a series of steps, where each step adheres to algebraic rules.
Let's look at these rules in detail:

• Multiplying or dividing both sides by the same value doesn't change the equality.
• Adding or subtracting the same amount from both sides is also allowed.

Using these manipulations allows for solving more complex formulas systematically when isolating the desired variable.

To isolate a variable means getting that variable alone on one side of the equation. This often involves reversing the operations done to it. Some steps might be:

• Eliminating fractions by multiplying both sides by a common denominator.
• Moving other terms to the opposite side by addition or subtraction.
• Undoing multiplication or division by performing the inverse operation.

In our specific example:

1. We start with: \(h = \frac{2 \, \mathscr{A}}{B + b}\).
2. Multiply both sides by \(B + b\) to remove the fraction: \(h(B + b) = 2 \mathscr{A}\).
3. Divide both sides by 2 to isolate \(\mathscr{A}\): \(\mathscr{A} = \frac{h(B + b)}{2}\).
These steps ensure that \(\mathscr{A}\) is now isolated on one side of the equation.

Rearranging equations involves changing the presentation of an equation for easier interpretation or to solve for a specific variable. The process requires correct application of algebraic manipulation.

In the provided example, we had to rearrange to solve for \(\mathscr{A}\). Here’s the detailed rearrangement:

1. Start with the given formula: \(h = \frac{2 \, \mathscr{A}}{B + b}\).
2. Multiply both sides by \(B + b\): \(h(B + b) = 2 \mathscr{A}\).
3. Divide both sides by 2 to isolate \(\mathscr{A}\): \(\mathscr{A} = \frac{h(B + b)}{2}\).
Rearranging the equation properly helps to present \(\mathscr{A}\) in a clear and usable form. Practice rearranging equations with different variables to become proficient.