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When working with algebraic fractions, finding the least common denominator (LCD) is an essential step to simplify the equation. The purpose of finding the LCD is to eliminate the fractions by combining them into a common base, making the equation easier to manipulate.

The least common denominator is essentially the smallest number or expression that all denominators can divide without leaving a remainder. In this specific problem, the denominators are variables, and the LCD becomes a product of these distinct terms: \(r \times r_{a} \times n_{0} \times r_{c}\).

By multiplying each term of the equation by this LCD, you transform each fraction into whole numbers, or algebraic terms, freeing the equation from fractions. This step simplifies the process of solving the equation as you're now working with standard algebraic expressions rather than fractions.

Once the fractions are cleared out by using the least common denominator, the next step is to simplify the resulting algebraic equation. Simplifying involves combining like terms and factoring whenever possible to reduce the equation to its simplest form.

In this exercise, after multiplying through by the LCD, you have an equation whose left side contains several components with common factors. The terms \(n_{0} \times r_{c} \times r\), \(r_{a} \times r_{c} \times r\), and \(r_{a} \times n_{0} \times r\) can be combined because they all involve the variable \(r\).

Group these terms and factor out \(r\), leading to a simplified expression: \(r \times (n_{0} \times r_{c} + r_{a} \times r_{c} + r_{a} \times n_{0}) = r_{a} \times n_{0} \times r_{c}\).

This simplification is crucial because it reduces the problem to a form where you can easily solve for \(r\) by isolating it on one side of the equation.

Solving rational equations involves isolating the variable you need to solve for, which is typically accomplished after simplifying the equation. In this case, the goal is to find the value of \(r\) that satisfies the entire equation.

Once you have the simplified equation \(r \times (n_{0} \times r_{c} + r_{a} \times r_{c} + r_{a} \times n_{0}) = r_{a} \times n_{0} \times r_{c}\), you can isolate \(r\) by dividing both sides by the expression that multiplies \(r\).

This results in the expression: \(r = \frac{r_{a} \times n_{0} \times r_{c}}{n_{0} \times r_{c} + r_{a} \times r_{c} + r_{a} \times n_{0}}\).

Dividing to solve for \(r\) is the final step in solving the equation. It shows how the values contained in the other variables affect the value of \(r\), giving a clear solution to the originally complex rational equation.