Solving Rational Equations And Identifying Restrictions
In the realm of mathematics, equations form the bedrock of problem-solving and analytical thinking. Among these, rational equations, characterized by the presence of variables in their denominators, present a unique set of challenges and considerations. This article delves into the intricacies of solving a specific rational equation, with a particular focus on identifying restrictions on the variable and subsequently determining the solution set. Our journey will involve a step-by-step exploration of the equation, emphasizing the critical concept of values that render the denominator zero, thereby imposing restrictions on the variable. Furthermore, we will navigate the algebraic manipulations necessary to isolate the variable and arrive at the solution, all while adhering to the fundamental principles of mathematical rigor.
The given equation is:
Identifying Restrictions on the Variable
In rational equations, the denominator plays a crucial role in defining the domain of the variable. A fundamental principle of mathematics dictates that division by zero is undefined. Consequently, any value of the variable that makes the denominator of a fraction equal to zero must be excluded from the solution set. These excluded values are termed restrictions on the variable.
To determine the restrictions for the given equation, we need to identify the values of x that would make the denominators 5x and 10x equal to zero.
For the denominator 5x, we set it equal to zero and solve for x:
5x = 0
Dividing both sides by 5, we get:
x = 0
Similarly, for the denominator 10x, we set it equal to zero and solve for x:
10x = 0
Dividing both sides by 10, we get:
x = 0
Therefore, the only value of x that makes either denominator zero is x = 0. This value must be excluded from the solution set, as it would lead to an undefined expression. Thus, the restriction on the variable is x ≠0.
This restriction is paramount in solving rational equations. Ignoring it can lead to extraneous solutions, which are values obtained algebraically but do not satisfy the original equation. Therefore, it is essential to identify and explicitly state the restrictions before proceeding with the solution process.
Solving the Rational Equation
Having identified the restriction on the variable, we can now proceed with solving the rational equation. The primary goal is to isolate the variable x on one side of the equation. To achieve this, we will employ a series of algebraic manipulations, ensuring that each step preserves the equality.
The equation is:
The first step is to eliminate the fractions. This can be done by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. The denominators are 5x, 4, 10x, and 3. The LCM of these denominators is 60x.
Multiplying both sides of the equation by 60x, we get:
60x() = 60x()
Distributing 60x on both sides, we have:
60x() + 60x() = 60x() - 60x()
Simplifying each term, we get:
24 + 15x = 234 - 20x
Now, we have a linear equation in x. To solve for x, we need to collect the terms with x on one side and the constant terms on the other side.
Adding 20x to both sides, we get:
24 + 15x + 20x = 234 - 20x + 20x
Simplifying, we have:
24 + 35x = 234
Subtracting 24 from both sides, we get:
24 + 35x - 24 = 234 - 24
Simplifying, we have:
35x = 210
Finally, dividing both sides by 35, we get:
Simplifying, we obtain:
x = 6
Verifying the Solution
It is crucial to verify the solution obtained against the restriction on the variable. We found that x = 6, and the restriction is x ≠0. Since 6 is not equal to 0, the solution satisfies the restriction.
To further ensure the correctness of the solution, we can substitute x = 6 back into the original equation and check if both sides are equal.
The original equation is:
Substituting x = 6, we get:
Simplifying, we have:
To add or subtract fractions, we need a common denominator. The LCM of 15, 4, 20, and 3 is 60. Converting each fraction to have a denominator of 60, we get:
Simplifying, we have:
Since both sides of the equation are equal, the solution x = 6 is verified.
Conclusion
In this comprehensive exploration, we successfully navigated the intricacies of solving a rational equation. We began by meticulously identifying the restrictions on the variable, emphasizing the critical concept of excluding values that would render the denominator zero. This step is paramount in ensuring the validity of the solution. Subsequently, we employed a series of algebraic manipulations to isolate the variable, arriving at the solution x = 6. Finally, we rigorously verified the solution against the restriction and by substituting it back into the original equation, confirming its correctness.
This exercise underscores the importance of a systematic approach to solving rational equations, with a keen focus on identifying restrictions, employing appropriate algebraic techniques, and verifying the solution. By adhering to these principles, we can confidently tackle a wide range of rational equations and unlock their solutions.
In summary, for the equation :
a. The value of the variable that makes a denominator zero is x = 0. This is the restriction on the variable.
b. The solution to the equation is x = 6.