When solving linear equations that contain fractions, the first step is to eliminate those fractions to simplify the equation. This can be achieved by using the least common denominator (LCD). For example, consider the equation:

\(\frac{1}{4}x + 2 - \frac{1}{3}x = \frac{1}{12}\).

In this case, the denominators are 4, 3, and 12, and the least common denominator is 12. To eliminate the fractions, multiply every term in the equation by 12:

\(12 \left(\frac{1}{4}x + 2\right) - 12 \left(\frac{1}{3}x\right) = 12 \left(\frac{1}{12}\right)\).

This simplifies to:

\(3x + 6 - 4x = 1\).

Now that the fractions are gone, the next step is to distribute any constants. Here, the 3 is distributed to both \(x\) and 2, resulting in:

\(3x + 6 - 4x = 1\).

Next, combine like terms. The terms \(3x\) and \(-4x\) combine to give \(-1x\), leading to:

\(-1x + 6 = 1\).

Now, isolate the variable \(x\) by moving the constant term to the other side. Subtract 6 from both sides:

\(-1x = 1 - 6\), which simplifies to \(-1x = -5\).

To solve for \(x\), divide both sides by -1:

\(x = \frac{-5}{-1} = 5\).

Finally, it’s a good practice to check your solution by substituting \(x\) back into the original equation. This step, while optional, helps confirm the accuracy of your solution.