91Ó°ÊÓ

When we talk about isolating the variable, we mean making the variable stand alone on one side of the equation. In this exercise, we aim to isolate the variable \( b \) in the equation \( P = \frac{1}{3} b h \).

To begin, observe the given equation and identify the variable you need to solve for. In this case, it's \( b \). Next, perform operations that will isolate the variable step by step.

Here, start by getting rid of the fraction. Multiply both sides of the equation by 3, resulting in:

\[ 3P = b h \]

Now the variable \( b \) is more straightforwardly part of a multiplication equation, making it easier to handle in future steps.

Mathematical operations like addition, subtraction, multiplication, and division are tools to manipulate equations. In this situation, we make use of multiplication and division.

\[ 3P = b h \]
To further isolate \( b \), we will divide both sides by \( h \), so the \( h \) on the right-hand side cancels out. This crucial step is called solving for the variable:

\[ b = \frac{3P}{h} \]
This final expression shows the variable \( b \) isolated. Always remember, any operation you perform on one side of the equation, you must also do on the other side. This ensures the equation remains balanced, much like a seesaw.

Verification is the final yet essential step to ensure your solutions are accurate. Take the equation given at the start: \( P = \frac{1}{3} b h \). Substitute the solution found back into the original equation:

Plug the value of \( b \) back into the equation:

\[ P \stackrel{?}{=} \frac{1}{3} \left( \frac{3P}{h} \right) h \]
Simplifying inside the parentheses first:

\[ \frac{1}{3} \cdot ( 3P \div h ) \cdot h = \frac{1}{3} \cdot \frac{3P \cdot h}{h} \]
Since the \( h \) terms cancel each other out, we are left with:

\[ P = P \]
This confirms that our solution \( b = \frac{3P}{h} \) is correct. Thus, verification reassures us that every arithmetic step taken holds true, solidifying our understanding and confidence in the solution.