91Ó°ÊÓ

When solving algebraic equations, one of the most important steps is isolating the variable you are solving for. In simple terms, this means getting the variable on its own on one side of the equation. For instance, in the equation given, \( g w^2 = 2r \), we want to solve for \( w \). The first step is to isolate the term \( w^2 \) by eliminating the coefficient \( g \). To do this, divide both sides of the equation by \( g \). This results in: \[ w^2 = \frac{2r}{g} \]This step is crucial because isolating the variable allows us to focus on the next part of solving the equation. Always remember, whatever you do to one side of the equation, you must do the same to the other side.

By mastering this step, you set a strong foundation for finding solutions to more complex algebraic problems.

Once the variable is isolated, the next step is to solve for the variable itself. In our problem, we have the equation: \[ w^2 = \frac{2r}{g} \]To solve for \( w \), we need to remove the exponent by taking the square root of both sides. This step involves understanding that taking the square root of a number can yield two results: a positive and a negative value. Mathematically, this is shown as: \[ w = \pm \sqrt{\frac{2r}{g}} \]This step is essential because it reflects the property that squaring either a positive or negative value will give the same result.

Keep in mind that while taking the square root, you are essentially undoing the squaring operation. Always denote the solution with both positive and negative roots to ensure all possible solutions are covered.

Algebraic formulas are fundamental to solving equations efficiently. They can be seen as tools that allow us to rearrange and solve for specific variables. In the exercise, the formula: \( g w^2 = 2r \)requires rearrangement to isolate \( w \). Understanding how to manipulate and rearrange algebraic formulas simplifies seemingly complex problems.

Let's review the process in simpler steps:

• Identify the term containing the variable of interest.
• Perform algebraic operations to isolate that term.
• Take further steps, like square root or division, to solve for the variable.

These steps show how versatile algebraic manipulation can be for solving a range of problems. Becoming familiar with applying algebraic formulas will give you confidence in tackling different equations and finding solutions more easily.

Practice these principles, and you will find that manipulating algebraic formulas becomes like second nature.