91Ó°ÊÓ

In solving linear equations, isolating the variable is one of the crucial first steps. In our exercise, the equation is \( \frac{x}{3} = 4 \). Our task is to solve for \( x \) by isolating it on one side of the equation.

To isolate a variable means to get it alone on one side of the equation, typically the left in most standard forms. In this example, \( x \) is currently divided by 3. To remove the division and isolate \( x \), you'll want to perform the inverse operation, which brings us to the next step. By isolating \( x \), you have effectively unraveled the puzzle, revealing its value.

Here is a simple process to help isolate variables:

• Identify what is happening to the variable. In this equation, \( x \) is divided by 3.
• Perform the inverse operation to both sides. Here, multiplying both sides by 3.

Once you've isolated \( x \) by multiplying both sides by 3, you get \( x = 12 \). This confirms that \( x \) is no longer part of a fraction, effectively isolating it.

Inverse operations are key to manipulating algebraic expressions and equations to isolate variables successfully. They are operations that reverse the effect of another operation.

For example, multiplication is the inverse operation of division. In our exercise, since \( x \) is divided by 3 in \( \frac{x}{3} = 4 \), multiplication by 3 is used to remove the division.

Here’s how inverse operations function:

• Identify the operation being performed on the variable. In this case, division by 3.
• Apply the inverse operation to both sides of the equation. So, multiply both sides by 3.

By applying these inverse operations, \( x \) is freed from the division by 3, leaving it as \( x = 12 \), which is the critical insight into finding the solution of the linear equation.

After isolating the variable and solving for its value, it's crucial to verify that the solution is correct by plugging it back into the original equation.

Verification serves as a double-check mechanism to ensure there are no errors in calculations or logical steps. Here's how you can verify an equation:

• Substitute the found value back into the original equation. For our exercise, substitute 12 for \( x \) into \( \frac{x}{3} \).
• Simplify the equation after substitution. You should get \( \frac{12}{3} = 4 \), which simplifies to 4 = 4, a true statement.

Since both sides of the equation are equal, you can confidently say the solution \( x = 12 \) is correct. Equation verification is instrumental in reinforcing the reliability of your solution approach.