Solving Rational Equations A Step By Step Guide
In this article, we will walk you through the process of solving the equation . This type of equation involves rational expressions, which are essentially fractions where the numerator and denominator are polynomials. Solving such equations requires careful attention to detail, particularly when dealing with potential extraneous solutions. Let's delve into the step-by-step method to tackle this problem effectively.
Step 1: Identify the Domain and Excluded Values
The first crucial step in solving any equation involving rational expressions is to identify the domain and any excluded values. The domain represents the set of all possible values of x for which the equation is defined. Excluded values are those values of x that would make the denominator of any fraction in the equation equal to zero, which is undefined in mathematics. These values must be excluded from the solution set. In our equation, the denominator is (x-3). To find the excluded values, we set the denominator equal to zero and solve for x:
x - 3 = 0 x = 3
Thus, x = 3 is an excluded value. This means that any solution we find that equals 3 must be discarded because it would make the original equation undefined. Identifying the domain and excluded values is a critical first step in solving rational equations, as it helps us avoid extraneous solutions later on. Extraneous solutions are values that we obtain during the solving process but do not satisfy the original equation. In this case, the domain is all real numbers except for x = 3, which we need to keep in mind as we proceed with solving the equation.
Step 2: Eliminate the Fractions
To eliminate the fractions in the equation, we multiply both sides of the equation by the least common denominator (LCD) of all the fractions present. In this case, we only have one denominator, which is (x-3). Therefore, the LCD is simply (x-3). Multiplying both sides of the equation by (x-3) will clear the fractions and allow us to work with a simpler equation:
(x-3) * [\frac{x}{x-3} + 7] = (x-3) * [\frac{3}{x-3}]
Now, we distribute (x-3) to each term on both sides of the equation:
(x-3) * \frac{x}{x-3} + (x-3) * 7 = (x-3) * \frac{3}{x-3}
This simplifies to:
x + 7(x-3) = 3
By multiplying both sides of the equation by the least common denominator, we've successfully eliminated the fractions. This step is essential in solving rational equations because it transforms the equation into a more manageable form, typically a linear or quadratic equation. In our case, we've obtained a linear equation, which is much easier to solve. Eliminating fractions simplifies the equation and reduces the risk of errors in subsequent steps. The ability to clear fractions effectively is a cornerstone of algebraic manipulation when dealing with rational expressions.
Step 3: Simplify and Solve the Resulting Equation
Now that we have eliminated the fractions, we need to simplify and solve the resulting equation. We start by distributing the 7 in the term 7(x-3):
x + 7x - 21 = 3
Next, we combine like terms on the left side of the equation:
8x - 21 = 3
To isolate the variable x, we add 21 to both sides of the equation:
8x = 3 + 21 8x = 24
Finally, we divide both sides by 8 to solve for x:
x = \frac{24}{8} x = 3
After simplifying and solving, we find that x = 3. However, it is crucial to remember the excluded value we identified in Step 1. We found that x = 3 makes the denominator of the original equation equal to zero, which means it cannot be a valid solution. When solving rational equations, this step of simplifying and solving is fundamental. We've transformed the complex equation into a simple linear equation, making it easier to isolate the variable. The key is to follow the order of operations and algebraic principles accurately. Nevertheless, we always need to cross-check our solutions with the initially identified excluded values to avoid errors.
Step 4: Check for Extraneous Solutions
As we identified in Step 1, x = 3 is an excluded value because it makes the denominator of the original equation equal to zero. In Step 3, we found that x = 3 is the solution to the simplified equation. This means that x = 3 is an extraneous solution. An extraneous solution is a value that satisfies the transformed equation but not the original equation. In the context of solving rational equations, it's imperative to check for extraneous solutions because the process of clearing fractions can sometimes introduce solutions that are not valid in the original equation.
Since x = 3 is an extraneous solution, it cannot be included in the solution set. Therefore, the original equation has no solution. Checking for extraneous solutions is a critical step in solving rational equations. By plugging our solutions back into the original equation, we ensure that they do not make any denominators zero or lead to undefined expressions. This verification process helps maintain the integrity of our solution and avoids errors. When dealing with rational expressions, remember that not every solution obtained from the simplified equation is necessarily a solution to the original equation.
Step 5: State the Solution Set
Since x = 3 is an extraneous solution and there are no other solutions, the solution set for the equation is empty. We can represent this using the empty set symbol, ā , or by stating that there is no solution. In summary, the original equation has no solution. When solving rational equations, the final step is to explicitly state the solution set. This ensures clarity and completeness in the answer. It's important to accurately reflect the results of our analysis, especially when we encounter scenarios where there are extraneous solutions or no solutions at all. Stating the solution set clearly communicates our final answer and reinforces the rigor of the solving rational equations process.
Therefore, the solution set is { }. (Empty set)
By systematically following these steps ā identifying the domain and excluded values, eliminating fractions, simplifying the equation, checking for extraneous solutions, and stating the solution set ā we can confidently solve equations involving rational expressions.
