91Ó°ÊÓ

Isolating the variable is a fundamental step when solving a linear equation. Our goal is to have the variable, let's say \( x \), by itself on one side of the equation. To achieve this, we need to remove any other terms or numbers on the same side as the variable.
For instance, in the equation \( 2x + 9 - \sqrt{3} = 3 \), you start by isolating \( 2x \). We achieve this by using inverse operations, commonly subtraction or addition, to cancel other terms.
For this equation, subtract \( 9 - \sqrt{3} \) from both sides. This doesn't change the equation because you're performing the same operation on both sides, preserving the equality.
By isolating \( 2x \), the equation becomes \( 2x = -6 + \sqrt{3} \). More intuitively, think of isolating the variable as "peeling away" layers until you're left with just the variable.

After isolating the variable, it's important to simplify the expression to make it easier to handle. Simplifying involves performing operations so that terms are combined correctly and unnecessary complexity is reduced.
Looking at our expression, we have moved from \( 2x + 9 - \sqrt{3} = 3 \) to \( 2x = -6 + \sqrt{3} \). Here simplification had already been partially achieved when subtracting \( 9 - \sqrt{3} \) from both sides of the equation:

• Combining numbers: By handling \( 9 - 9 \) we simplify the constant terms to \( 0 \).
• Handling square roots: \( \sqrt{3} \) remains since it's already in its simplest form.

Even though our example didn't involve complicated fractions or multiple terms, understanding how to simplify makes solving the equation straightforward. Think of simplifying as tidying up the equation so it becomes clear and manageable.

The final step in any linear equation is solving for the variable. After isolating the term with \( x \) and simplifying the equation, you can solve for \( x\). In our example, we have arrived at the simplified form \( 2x = -6 + \sqrt{3} \).
To proceed, divide both sides of the equation by \( 2 \). This step is important because it effectively "frees" \( x \) from the coefficient that is multiplying it.
Taking the equation apart step-by-step helps ensure accuracy:

• You divide the entire right-hand expression \( -6 + \sqrt{3} \) by \( 2 \).
• Right division involves ensuring each term is thoroughly accounted for during the division.

Thus, after dividing, the equation \( x = \frac{-6 + \sqrt{3}}{2} \) gives us the value for \( x \).
This step completes the process by providing the solution to the equation. Solving for the variable thus unlocks the answer within the equation, revealing what \( x \) stands for in this context.