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Isolating variables is a fundamental skill in solving equations. It essentially means getting the variable you are solving for, usually represented as \( x \), alone on one side of the equation. This process involves reversing operations that are being applied to the variable. Think of it as peeling away layers until you reach the core value you need.
One technique often used is to apply the opposite operation. For instance, if a variable is divided by a number, as is the case in the equation \( \frac{x}{3} = 10 \), the inverse operation (multiplication) can be used to remove the division. By multiplying both sides of the equation by 3, an operation reverse to the division, you pull \( x \) out from under the division. This leaves you with the equation: \( x = 30 \).
Always remember, the operation applied to remove other terms around the variable should be consistent on both sides of the equation. This ensures the equation's balance is maintained and leads to the correct solution.

Once you've obtained a solution, like \( x = 30 \) in the example provided, it's crucial to verify this solution to ensure it's correct. This verification comes from substituting the solution back into the original equation. By doing this, you confirm whether both sides of the equation balance.
In our example, we replace \( x \) with 30 in \( \frac{x}{3} = 10 \). Calculating \( \frac{30}{3} \) results in 10, which indeed equals the right side of the equation. This balance confirms that \( x = 30 \) is the correct solution.
This step is vital, especially in more complex equations, as it helps catch errors that may have occurred during calculations. It's a simple yet effective check to ensure your work is accurate.

Multiplication is a key operation in rearranging equations, particularly for eliminating fractions or dividing terms. When you multiply both sides of an equation by the same number, you help maintain the equality, making it beneficial for isolating variables.
Consider the given example again: \( \frac{x}{3} = 10 \). Multiplying both sides by 3 effectively cancels out the division by 3 on the left side because \( 3 \times \frac{x}{3} \) simplifies directly to \( x \).
This method is straightforward but powerful, especially when dealing with equations that have fractions. Remember, consistency in multiplying both sides by the same number keeps the equation fair and balanced. This consistency is a foundational principle for solving any algebraic equation.