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When tackling rational equations, the first step is always to understand what you're working with. Rational equations are those that contain fractions with polynomials in their numerators or denominators. In this case, the equation involves simple fractions and looks like \( \frac{5}{x} + \frac{1}{3} = \frac{6}{x} \). Our aim is to find the value of \(x\) that makes this equation true.
Solving involves rearranging and simplifying parts of the equation until you isolate the variable (\(x\) in this case). By doing this systematically, you will avoid mistakes and arrive at the correct solution. This is particularly important in math because even a small error can lead to incorrect conclusions.

Combining like terms is an essential strategy in simplifying equations. In the equation \( \frac{5}{x} + \frac{1}{3} = \frac{6}{x} \), you can see that the terms \( \frac{5}{x} \) and \( \frac{6}{x} \) each have \(x\) as their denominator.
To combine them, you subtract \( \frac{6}{x} \) from \( \frac{5}{x} \). This gives \( \frac{5}{x} - \frac{6}{x} = \frac{-1}{x} \). By consolidating these terms, the equation becomes simpler, paving the way to the next steps.
Be vigilant while doing this step as combining like terms correctly will set the foundation for the upcoming fraction elimination.

Eliminating fractions is crucial when dealing with rational equations, as it simplifies the expression and makes solving easier. Here, once we've combined the terms to \( \frac{-1}{x} = -\frac{1}{3} \), the next logical step is to eliminate the fractions.
To do this, multiply every term in the equation by a common denominator, which in our simplified equation's case is \(-3x\). This multiplication cancels out the denominators and leaves us with \(x = -3\).
This is a critical juncture in solving rational equations, ensuring no fraction remains, thus making the variable more accessible.

Once a solution is obtained, verifying it ensures accuracy and reliability. To verify the solution \(x = -3\), substitute \(-3\) back into the original equation and check whether both sides equate.
Substituting, you have: \( \frac{5}{-3} + \frac{1}{3} = \frac{6}{-3} \). Simplifying both sides, \(-\frac{5}{3} + \frac{1}{3} = -2\). Compute both sides; indeed, they equal \(-2\), matching perfectly.
Verification is a vital step in mathematics, giving confidence that the solution is correct. It acts as a safety net, capturing any possible mishaps that may have occurred during the solving process.