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When solving rational equations, cross-multiplication is a key technique to eliminate fractions. This is particularly useful in problems where two fractions are set equal. In this method, you multiply the numerator of one fraction by the denominator of the other, and vice versa. For example, given the equation \(\frac{n+3}{n-5}=\frac{1}{2}\), we use cross-multiplication to obtain \[ 2(n + 3) = 1(n - 5) \]. This converts the equation into a simpler linear form that can be solved without fractions.

Cross-multiplication makes it easier to handle fractions by transforming them into regular algebraic equations. When applying this technique, remember to multiply correctly and simplify the equation step-by-step.

Isolating the variable is an essential practice to solve for it in equations. After using cross-multiplication, the next step is to rearrange the equation to isolate the variable. In our example, we have the simplified equation \[ 2n + 6 = n - 5 \].

To isolate \(n\), follow these steps:

• Subtract \(n\) from both sides: \[ 2n - n + 6 = -5 \] \Rightarrow \[ n + 6 = -5 \]
• Subtract 6 from both sides: \[ n + 6 - 6 = -5 - 6 \] \Rightarrow \[ n = -11 \]

Step by step, reposition the terms involving \(n\) on one side and constants on the other. This process 'solves' the equation by revealing the value of the unknown variable.

Always verify your solutions to ensure they are correct. For rational equations, substitute the found solution back into the original equation to check its validity. In our case, we found \(n = -11\).

Substitute \(n = -11\) into the original equation:

• Original equation: \(\frac{n+3}{n-5}=\frac{1}{2}\)
• Substitution: \(\frac{-11+3}{-11-5}=\frac{1}{2}\)

This simplifies to \(\frac{-8}{-16}=\frac{1}{2}\), which is true since \(\frac{-8}{-16} = \frac{1}{2}\).

If the left-hand side equals the right-hand side after substitution, the solution is verified. Verifying not only confirms correctness but also builds confidence in your solving process.