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Isolating a variable is a fundamental skill in solving equations. It involves rearranging an equation so that the variable you are solving for stands alone on one side of the equation. This helps in simplifying and understanding the relationship between variables and constants within an equation.
To start, identify the variable you want to isolate. Look at its coefficients, which are numbers multiplying the variable, and figure out how to eliminate them from one side of the equation.

• This often means subtracting or adding terms from both sides.
• You might also need to multiply or divide to fully isolate the variable.

For example, in the equation \(\frac{x}{2} + 4 = \frac{3x}{2}\), you need to first move all terms containing \(x\) to one side. This was done by subtracting \(\frac{x}{2}\) from both sides to simplify to \(4=\frac{x}{2}\).
Always remember, whatever operation you perform on one side of the equation, you must do the same on the other side to maintain the equality.

Linear equations are equations where the highest power of the variable is one. They are the simplest form of equations and form straight lines when graphed.
These equations can take different forms, but they usually resemble one of the following templates:

• Standard form: \(Ax + By = C\)
• Slope-intercept form: \(y = mx + b\)

In our exercise, the equation \(\frac{x}{2} + 4 = \frac{3x}{2}\) is a linear equation. It's expressed in terms of \(x\) and constants.
The key objective with linear equations is often to solve for the specific value of the variable that makes the equation true. For the above example, isolating and simplifying gives us \(x = 8\).

Fractional equations involve variables within fractions, creating an extra challenge compared to standard linear equations. The principle is still the same: isolate the variable while simplifying the equation.
Key steps involve:

• Clearing the fraction by finding a common denominator.
• Multiplying through by this denominator to eliminate fractions.

In the exercise \(\frac{x}{2} + 4 = \frac{3x}{2}\), both sides of the equation have fractions. By subtracting \(\frac{x}{2}\) from \(\frac{3x}{2}\), we simplified it to \(4 = \frac{x}{2}\).
Next, we eliminated the fraction by multiplying the entire equation by 2, resulting in \(x=8\). Keep in mind that this multiplication step is crucial for removing fractions and simplifying your path to the solution.