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Isolating the variable means getting the variable term by itself on one side of the equation. This often requires multiple steps, such as adding, subtracting, multiplying, or dividing terms. In the initial steps of isolating the variable, we want to group any like terms that involve the variable.

For example, consider the equation: \ \frac{3}{2} x + 2 = \frac{1}{2} - \frac{1}{2} x \ The goal is to have all terms with \( x \) on one side. We do this by adding \ \frac{1}{2} x \ to both sides:

\( \frac{3}{2} x + 2 + \frac{1}{2} x = \frac{1}{2} - \frac{1}{2} x + \frac{1}{2} x \)

This simplifies the equation to: \( 2x + 2 = \frac{1}{2} \)

Now the next step is to subtract 2 from both sides to isolate the variable term completely:

\( 2x + 2 - 2 = \frac{1}{2} - 2 \)\( 2x = \frac{1}{2} - 2 \)

Simplifying equations helps make them manageable and easier to solve. The process often involves combining like terms, reducing fractions, or converting terms into simpler forms. If you isolate the variable, you must bring simplicity into the equation by reducing it step-by-step.

From our previous steps, we reached:

\( 2x = \frac{1}{2} - 2 \)

Convert \ 2 \ to a fraction with the same denominator:

\( 2 = \frac{4}{2} \)

Now, subtract \ \frac{4}{2} \ from \ \frac{1}{2} \:

\( 2x = \frac{1}{2} - \frac{4}{2} \)

This simplifies to:

\( 2x = -\frac{3}{2} \)

By combining and reducing terms, our equations simplify as we progress through the steps. This makes solving for the variable straightforward and less error-prone.

The final goal is to solve for the variable \( x \), which means finding its value. This usually comes after isolating the variable and simplifying the equation.

At this stage in our example, we have:

\( 2x = -\frac{3}{2} \)

We solve for \( x \) by dividing both sides by the coefficient of \( x \), which is 2:

\( \frac{2x}{2} = \frac{-\frac{3}{2}}{2} \)

Reducing this, we get:

\( x = -\frac{3}{4} \)

Thus, \( x = -\frac{3}{4} \). This means we've solved the equation and found the value of \( x \). Each step methodically leads us to this final answer. Reviewing these steps ensures we understand the entire process and can apply it to similar problems.