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Cross-multiplication is a popular method used to eliminate fractions in rational equations, transforming equations into a more straightforward form. This technique relies on the fundamental property that any fraction \can be represented as two sets of products that are equal, as long as you multiply across the equal sign. For the given original exercise \\( \frac{2}{y} + \frac{1}{2} = \frac{5}{2y} \), \we apply cross-multiplication to clear out the fractions. \

This method involves multiplying each term by the common denominators present in the equation. Here, the common denominator is \\(2y\). By multiplying every term in the equation by \\(2y\), \the fractions are eliminated, which results in an equation without fractions: \\(4 + y = 5\). \

Using cross-multiplication can simplify a complex looking rational equation significantly. This prepares you easily for the following steps of solving for the variable.

Once you believe you've found a solution to your equation, such as in the step-by-step process where we determined \\(y = -1\), \it's crucial to check this value against the original equation. To do this, replace the variable in the original equation with the solution and verify whether the left-hand side equal the right-hand side. \

In our example: \\(\frac{2}{-1} + \frac{1}{2} = \frac{5}{2*-1}\), \which simplifies to \\(-2 + 0.5 = -2.5\), \was found incorrect. \This is because \\(-1.5 eq -2.5\). \

Checking solutions is not just about confirming answers; it helps identify any error in calculation steps, prompting a re-evaluation. This ensures the answer is both accurate and valid, reinforcing a strong mathematical routine.

Simplifying equations is a key step in solving rational equations as it makes handling solutions approachable and less tedious. \In our specific case, after employing cross-multiplication, we reached an equation without fractions: \\(4 + y = 5\).Simplifying further by isolating the variable \helps find the answer. \

To solve for \\(y\), \subtract \\(y\) \from both sides to eliminate the variable on one side: \\(4 + y - y = 5 - y\) \which becomes \\(4 = 5 - y\). \Next, simplifying by subtracting \\(5\) from both sides \yields: \\(-1 = -y\), \which allows us to find \\(y = -1\). \

Despite initially incorrect results in checking due to calculation errors, it's clear that correct simplification leads to the correct solution. Simplification ensures clarity and straightforward paths to answers, and when paired with solution checking, it's an effective strategy in problem solving.