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When dealing with rational equations, finding a common denominator is crucial to simplifying and solving the equation. Rational equations are equations that involve fractions and usually have different denominators. By transforming all terms to have the same denominator, you can more easily manipulate and combine these fractions. In our problem, the denominators are originally \(6n - 6\) and \(4n - 4\). By factoring, these become \(6(n - 1)\) and \(4(n - 1)\), respectively. To find the least common denominator, you consider the highest power of each factor present in these expressions.

But why is this important? Having a common denominator allows us to combine fractions in a single expression. In practical terms, it's like combining like terms in algebra; the fractions can be added or subtracted seamlessly. Here, the least common denominator is \(12(n - 1)\). Transforming each fraction in the equation to have this common denominator is a critical step in simplifying the rational equation before solving it.

Excluded values in rational equations are specific values of the variable that would make any denominator zero. Since division by zero is undefined, these values must be removed from the solution set. To identify excluded values, examine each denominator in the equation. In our exercise, the denominators are \(6n - 6\) and \(4n - 4\). Set each equal to zero and solve:

• \(6n - 6 = 0\) implies \(n = 1\)
• \(4n - 4 = 0\) also implies \(n = 1\)

Therefore, \(n = 1\) is excluded from the domain of the solution.

It's essential to check your solution against these excluded values after solving the equation to ensure validity. In practice, it prevents the occurrence of mathematical errors that don't make logical sense. For our equation, we verified that the found solution, \(n = \frac{22}{5}\), is not an excluded value, confirming its validity.

Once you've identified a common denominator and rewritten the equation, the next step is to solve the resulting linear equation. A linear equation is an equation of the first degree, meaning it has no exponents higher than one and typically looks like \(ax + b = c\). After rewriting the rational equation with a common denominator, equate the numerators. In our example, you end up with the equation \(2(n - 5) = 12 - 3n\). The goal is to isolate \(n\).

Follow these steps:

• Expand and simplify each side. \(2n - 10 = 12 - 3n\).
• Add \(3n\) to both sides to eliminate the \(-3n\) from the right: \(5n - 10 = 12\).
• Add 10 to both sides: \(5n = 22\).
• Divide by 5 to solve for \(n\): \(n = \frac{22}{5}\).

Solving the equation correctly leads to the solution, providing you an answer that is consistent and accurate, avoiding the excluded values previously identified. Linear equations are foundational, and mastering them is an essential skill for successfully solving rational equations.