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When solving linear equations like \(\frac{1}{3} m-1=5\), the key is to isolate the variable, which means getting the variable on one side of the equation all by itself. The aim is to simplify the equation to a form where the variable, in this case 'm', stands alone and equals a numerical value on the other side of the equation.

To do this, you'll need to perform operations that reverse what's currently being applied to the variable. If the variable is being subtracted by 1, as in our initial equation, then adding 1 to both sides will counteract this subtraction. This is a crucial step in simplifying the equation and preparing it for further operations which will ultimately lead you to the solution.

Encountering fractions in equations may seem daunting, but they obey the same principles as whole numbers. For the equation \(\frac{1}{3} m-1=5\), after isolating the fraction \(\frac{1}{3}m\) on one side, you want to eliminate the fraction to solve for 'm'.

To get rid of the fraction, multiply both sides of the equation by the denominator of the fraction, which is 3 in this case. This step, called 'clearing the fraction', gives us \(m = 6\times 3\), and is intended to simplify the equation further into an integer-to-integer relationship, a simpler form for most students to handle and understand.

After obtaining the solution \(m = 18\), a crucial step is to verify the correctness of your solution. This process, known as equation checking, involves substituting the solution back into the original equation to ensure the resulting statement is true.

Plugging \(m = 18\) back into \(\frac{1}{3} m-1=5\) results in \(\frac{1}{3}\times 18 - 1 = 5\), which simplifies to \(6 - 1 = 5\). This confirms that the original equation holds true for \(m = 18\), thereby validating our solution. Making a habit of checking your solutions can prevent errors and ensure a deeper understanding of the mathematical principles at work.