Solving 1/(c-3) - 1/c = 3/[c(c-3)] Finding The Solution Set

In this article, we will delve into the solution of the rational equation $\frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)}$. This equation, involving fractions with polynomials in the denominator, requires a careful approach to identify any potential solutions while also considering restrictions on the variable c. We will explore the steps involved in simplifying the equation, finding a common denominator, and solving for c. Additionally, we will pay close attention to values of c that would make the denominators zero, as these values are excluded from the solution set. Our goal is to provide a comprehensive understanding of how to tackle such equations and to arrive at the correct solution.

Rational equations are equations that contain rational expressions, which are essentially fractions with polynomials in the numerator and/or denominator. Solving these equations involves algebraic manipulations to eliminate the fractions and arrive at a simpler equation that can be solved using standard techniques. However, a crucial aspect of solving rational equations is identifying any values of the variable that would make the denominator zero. These values are not part of the solution set because division by zero is undefined. Before we dive into solving our specific equation, let's establish the groundwork for understanding rational equations. Rational equations, at their core, involve fractions where the numerator and/or the denominator contain variables. The primary strategy for solving these equations is to eliminate the fractions by finding a common denominator and multiplying both sides of the equation by it. This transforms the equation into a more manageable form, often a polynomial equation. However, this process introduces a critical consideration: extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation. They arise when we multiply both sides by an expression that can be zero for certain values of the variable. Therefore, it's paramount to check all potential solutions in the original equation to ensure they are valid. Furthermore, the domain of a rational equation is restricted by any values that make the denominator zero. These values are excluded from the solution set. For instance, in our equation, $\frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)}$, we immediately see that c cannot be 0 or 3. These restrictions are essential to keep in mind throughout the solving process. Understanding these principles is fundamental to accurately solving rational equations and avoiding common pitfalls. With a solid grasp of these concepts, we can confidently approach the given equation and determine its solution set.

Solving the Equation: A Step-by-Step Approach

To solve the given equation, $\frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)}$, we follow a systematic approach. First, we identify the restrictions on the variable c. As mentioned earlier, c cannot be 0 or 3, as these values would make the denominators zero. Next, we find a common denominator for the fractions. The common denominator in this case is c( c - 3). We then multiply both sides of the equation by this common denominator to eliminate the fractions. This gives us:

c( c - 3) * [$\frac{1}{c-3} - \frac{1}{c}$] = c( c - 3) * [$\frac{3}{c(c-3)}$]

Distributing c( c - 3) on the left side, we get:

c - (c - 3) = 3

Simplifying the left side, we have:

c - c + 3 = 3

Which further simplifies to:

3 = 3

This result indicates an identity, meaning the equation is true for all values of c that are within the domain. However, we must remember the restrictions we identified at the beginning. Since c cannot be 0 or 3, these values are excluded from the solution set. Therefore, while the equation holds true for most values of c, the solution must exclude c = 0 and c = 3 due to the original equation's denominators. The solution process highlights the importance of carefully considering restrictions when dealing with rational equations. Failing to account for these restrictions can lead to incorrect conclusions about the solution set. In summary, the equation 3 = 3 is an identity, but the solution to the original equation is all real numbers except c = 0 and c = 3.

Detailed Steps for Solving the Rational Equation

Let's meticulously walk through each step involved in solving the rational equation $\frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)}$. This detailed breakdown will reinforce the concepts and techniques used in solving such equations. Step 1: Identify Restrictions. The first crucial step is to identify any values of c that would make the denominators in the equation equal to zero. This is because division by zero is undefined, and these values must be excluded from the solution set. In our equation, we have denominators of c - 3, c, and c( c - 3). Setting each of these equal to zero, we find:

• c - 3 = 0 => c = 3
• c = 0
• c( c - 3) = 0 => c = 0 or c = 3

Thus, c cannot be 0 or 3. These are the restrictions on our solution. Step 2: Find the Common Denominator. To combine the fractions on the left side of the equation, we need a common denominator. The common denominator for c - 3 and c is their product, which is c( c - 3). Notice that this is also the denominator on the right side of the equation. Step 3: Rewrite the Fractions with the Common Denominator. We rewrite each fraction with the common denominator:

• $\frac{1}{c-3}$ becomes $\frac{c}{c(c-3)}$
• $\frac{1}{c}$ becomes $\frac{c-3}{c(c-3)}$

Now, the equation becomes:

$\frac{c}{c(c-3)} - \frac{c-3}{c(c-3)} = \frac{3}{c(c-3)}$

Step 4: Combine the Fractions. We can now combine the fractions on the left side:

$\frac{c - (c-3)}{c(c-3)} = \frac{3}{c(c-3)}$

Simplifying the numerator:

$\frac{3}{c(c-3)} = \frac{3}{c(c-3)}$

Step 5: Multiply Both Sides by the Common Denominator. To eliminate the fractions, we multiply both sides of the equation by c( c - 3):

c( c - 3) * $\frac{3}{c(c-3)}$ = c( c - 3) * $\frac{3}{c(c-3)}$

This simplifies to:

3 = 3

Step 6: Interpret the Result. The equation 3 = 3 is an identity, which means it is true for all values of c. However, we must consider the restrictions we identified in Step 1. Since c cannot be 0 or 3, the solution set includes all real numbers except 0 and 3. This detailed step-by-step solution provides a clear understanding of the process and highlights the importance of each step in solving rational equations. By carefully following these steps, we can accurately determine the solution set while avoiding common errors.

Identifying Restrictions on the Variable

Identifying restrictions on the variable is a critical step when solving rational equations. These restrictions stem from the fundamental mathematical principle that division by zero is undefined. In the context of rational equations, this means that any value of the variable that makes a denominator equal to zero must be excluded from the solution set. Failing to identify and account for these restrictions can lead to extraneous solutions or an incorrect understanding of the equation's solution. To effectively identify restrictions, one must examine each denominator in the equation and determine the values of the variable that would cause it to become zero. For instance, in the equation $\frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)}$, we have three denominators to consider: c - 3, c, and c( c - 3). Setting each of these expressions equal to zero allows us to find the restricted values. For c - 3 = 0, we add 3 to both sides, resulting in c = 3. This means that c cannot be 3, as this would make the denominator c - 3 equal to zero. Similarly, for c = 0, it is immediately clear that c cannot be 0. Lastly, for c( c - 3) = 0, we have two factors that could potentially be zero: c and c - 3. As we've already established, c cannot be 0 or 3. Therefore, the restrictions on the variable c in this equation are c ≠ 0 and c ≠ 3. These restrictions define the domain of the equation, which is the set of all possible values of c for which the equation is defined. Understanding the domain is crucial for interpreting the solution correctly. Once we solve the equation, we must ensure that our solutions fall within the domain. Any potential solutions that violate the restrictions are extraneous and must be discarded. This rigorous approach to identifying and applying restrictions ensures the accuracy and validity of the solution to the rational equation.

Common Mistakes to Avoid

When solving rational equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls is crucial for accurately solving these types of equations. One of the most frequent errors is failing to identify and account for restrictions on the variable. As previously discussed, values that make the denominator zero must be excluded from the solution set. Overlooking this step can lead to the inclusion of extraneous solutions, which satisfy the transformed equation but not the original one. Another common mistake is incorrectly applying the distributive property when multiplying both sides of the equation by the common denominator. It is essential to ensure that every term on both sides of the equation is multiplied by the common denominator. Neglecting to do so can alter the equation and result in an incorrect solution. A further pitfall lies in simplifying the equation prematurely. Before multiplying by the common denominator, it's best to combine fractions on each side of the equation, if possible. This reduces the complexity of the equation and minimizes the chance of errors. Additionally, it is crucial to check all potential solutions in the original equation. This step verifies that the solutions are valid and not extraneous. Substituting the solutions back into the original equation helps confirm that they do not result in division by zero or any other inconsistencies. Finally, errors can arise from algebraic mistakes during simplification or solving the resulting equation. Careful attention to detail and double-checking each step can help prevent these errors. By being mindful of these common mistakes and taking the necessary precautions, students can improve their accuracy and confidence in solving rational equations. A systematic approach, combined with a thorough understanding of the underlying principles, is key to success in this area of algebra.

In conclusion, the solution to the equation $\frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)}$ is all real numbers except c = 0 and c = 3. This solution arises from the fact that the equation simplifies to an identity (3 = 3), which is true for all values of c. However, the original equation has restrictions due to the denominators, which cannot be zero. Therefore, we must exclude c = 0 and c = 3 from the solution set. This exercise demonstrates the importance of not only solving equations correctly but also considering the context and restrictions imposed by the equation's structure. Understanding the domain of rational expressions is crucial for avoiding extraneous solutions and accurately interpreting the results. The step-by-step approach outlined in this article provides a solid framework for tackling similar rational equations. By carefully identifying restrictions, finding common denominators, and checking solutions, one can confidently navigate these types of problems and arrive at the correct answer. The key takeaway is that solving rational equations requires a combination of algebraic manipulation and a thorough understanding of the underlying mathematical principles.