Solving The Equation 2/x + 2 = 2/(3x) + 13/6 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving a specific equation: $ rac{2}{x}+2= rac{2}{3 x}+ rac{13}{6}$. We will explore the steps involved, providing a detailed explanation to enhance your understanding.
1. Understanding the Equation
Before diving into the solution, it's crucial to understand the equation. The equation $ rac{2}{x}+2= rac{2}{3 x}+ rac{13}{6}$ is a rational equation, which means it involves fractions with variables in the denominator. Our goal is to isolate the variable x to find its value(s) that satisfy the equation. To effectively tackle this equation, it's essential to grasp the concept of rational equations and the techniques used to solve them. A rational equation is essentially an equation that contains at least one fraction whose numerator and denominator are polynomials. These types of equations often require a bit more algebraic manipulation than simpler linear or quadratic equations. The primary strategy for solving rational equations revolves around eliminating the fractions. This is typically achieved by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions involved. The LCD is the smallest multiple that all the denominators divide into evenly. This step clears the fractions, transforming the equation into a more manageable form, usually a linear or quadratic equation, which can then be solved using standard algebraic techniques. However, a crucial step in solving rational equations is checking for extraneous solutions. Since we are dealing with variables in the denominators, some solutions obtained algebraically might make the denominator zero, which is undefined. These are called extraneous solutions and must be discarded. Therefore, after finding potential solutions, it is necessary to substitute them back into the original equation to ensure they do not result in division by zero. This verification process is vital to ensure the correctness of the solution. Once we have identified and discarded any extraneous solutions, the remaining solutions are the valid solutions to the original rational equation. In summary, solving rational equations involves clearing the fractions by multiplying by the LCD, solving the resulting equation, and then carefully checking for extraneous solutions. This methodical approach ensures accurate solutions and a solid understanding of rational equations.
2. Finding the Least Common Denominator (LCD)
To eliminate the fractions, we need to find the least common denominator (LCD) of the denominators x, 3x, and 6. The LCD is the smallest multiple that all the denominators divide into evenly. In this case, the LCD is 6x. Understanding how to find the least common denominator (LCD) is a cornerstone of working with fractions, especially when solving equations that involve them. The LCD is the smallest multiple that all the denominators in a set of fractions can divide into without leaving a remainder. This concept is not just limited to numerical fractions; it extends to algebraic fractions as well, where the denominators may contain variables. To find the LCD, one common method is to first list the multiples of each denominator until a common multiple is found. However, for more complex denominators, a more systematic approach involves prime factorization. Each denominator is broken down into its prime factors, and the LCD is constructed by taking the highest power of each prime factor that appears in any of the denominators. For example, if the denominators are 12 and 18, their prime factorizations are $2^2 * 3$ and $2 * 3^2$, respectively. The LCD is then $2^2 * 3^2 = 36$. In the context of algebraic fractions, the process is similar but involves factoring polynomials as well. For instance, if the denominators are x and x+1, the LCD is simply x(x+1) since they have no common factors. If the denominators are $x^2 - 4$ and x - 2, we would factor $x^2 - 4$ as (x-2)(x+2), and the LCD would be (x-2)(x+2) since x - 2 is a factor of $x^2 - 4$. Identifying the LCD is a critical step because it allows us to combine or eliminate fractions in equations or expressions. When solving equations, multiplying both sides by the LCD clears the denominators, transforming the equation into a more manageable form, such as a linear or quadratic equation. In adding or subtracting fractions, expressing each fraction with the LCD as the denominator allows for the numerators to be combined. The LCD is also crucial in simplifying complex fractions, where a fraction contains fractions in its numerator, denominator, or both. By multiplying the numerator and denominator of the complex fraction by the LCD of all the fractions within it, the complex fraction can be simplified into a single fraction. Therefore, mastering the technique of finding the LCD is essential for success in algebra and beyond, providing a solid foundation for more advanced mathematical concepts and applications.
3. Multiplying by the LCD
Multiply both sides of the equation by the LCD, 6x:
6x * ( rac{2}{x}+2) = 6x * ( rac{2}{3 x}+ rac{13}{6})
This step eliminates the fractions:
Multiplying both sides of an equation by the least common denominator (LCD) is a fundamental technique in algebra, especially when dealing with rational equations or equations involving fractions. This process effectively clears the fractions from the equation, transforming it into a more manageable form, typically a linear or quadratic equation. The underlying principle behind this technique is the multiplicative property of equality, which states that if you multiply both sides of an equation by the same non-zero quantity, the equality remains true. By choosing the LCD as the multiplier, we ensure that each denominator in the original equation will divide evenly into the LCD. This results in the fractions being simplified to whole numbers or polynomials, thus eliminating the fractions. The process starts with identifying the LCD of all the fractions present in the equation. As discussed previously, this involves finding the smallest multiple that all the denominators can divide into without leaving a remainder. Once the LCD is determined, both sides of the equation are multiplied by it. This multiplication must be distributed to each term on both sides of the equation to maintain balance. After distributing the LCD, each fraction is simplified by canceling out common factors between the denominators and the LCD. This step is where the fractions are effectively eliminated, leaving an equation that is easier to solve. For example, consider an equation like $ rac{x}{2} + rac{1}{3} = rac{5}{6}$. The LCD of 2, 3, and 6 is 6. Multiplying both sides of the equation by 6 gives $6( rac{x}{2} + rac{1}{3}) = 6( rac{5}{6})$. Distributing the 6 on both sides yields $6 * rac{x}{2} + 6 * rac{1}{3} = 6 * rac{5}{6}$, which simplifies to 3x + 2 = 5. Now, the equation is free of fractions and can be easily solved for x. In more complex equations involving polynomials in the denominators, the process is similar but may require factoring to determine the LCD and to simplify after multiplication. The key is to ensure that the LCD is correctly identified and that the multiplication is distributed properly. This technique is not only useful for solving equations but also for simplifying expressions and performing other algebraic manipulations. By eliminating fractions, the expressions become easier to work with, making subsequent steps more straightforward. Therefore, mastering the technique of multiplying by the LCD is crucial for success in algebra and provides a solid foundation for more advanced mathematical concepts.
4. Simplifying the Equation
Combine like terms:
Simplifying an equation is a crucial step in the process of solving it, as it involves reducing the equation to its simplest form while maintaining the equality. This process typically includes combining like terms, applying the distributive property, and performing basic arithmetic operations to both sides of the equation. The goal is to isolate the variable on one side of the equation, making it easier to determine its value. Combining like terms is a fundamental aspect of simplifying equations. Like terms are terms that contain the same variable raised to the same power. For example, 3x and 5x are like terms, while 3x and $3x^2$ are not. To combine like terms, we simply add or subtract their coefficients. For instance, 3x + 5x simplifies to 8x. Similarly, constant terms can be combined; for example, 7 - 4 simplifies to 3. The distributive property is another essential tool for simplifying equations, especially when dealing with expressions in parentheses. The distributive property states that a(b + c) = ab + ac. In other words, the term outside the parentheses is multiplied by each term inside the parentheses. For example, 2(x + 3) simplifies to 2x + 6. Applying the distributive property helps to remove parentheses and combine terms that were previously separated. Once like terms have been combined and the distributive property has been applied, the equation can be further simplified by performing basic arithmetic operations. This might involve adding, subtracting, multiplying, or dividing constants or terms on both sides of the equation to maintain equality. For example, if we have the equation 2x + 5 = 11, we can subtract 5 from both sides to get 2x = 6. Then, we can divide both sides by 2 to isolate x, resulting in x = 3. In more complex equations, the simplification process may involve multiple steps and the application of several algebraic properties. For instance, in an equation like $3(x + 2) - 2x = 7$, we would first apply the distributive property to get 3x + 6 - 2x = 7. Then, we would combine like terms to get x + 6 = 7. Finally, we would subtract 6 from both sides to isolate x, resulting in x = 1. Simplification not only makes equations easier to solve but also reduces the chances of making errors in subsequent steps. By carefully applying algebraic principles and performing operations methodically, we can transform complex equations into simpler, more manageable forms. This skill is crucial for success in algebra and provides a strong foundation for tackling more advanced mathematical problems.
5. Solving for x
Divide both sides by -1:
Therefore, the solution to the equation is x = 8. Solving for x is the ultimate goal in many algebraic problems, as it involves isolating the variable x on one side of the equation to determine its value. This process typically involves a series of algebraic manipulations, including applying inverse operations, combining like terms, and using properties of equality. The objective is to undo the operations that are being performed on x until it stands alone on one side of the equation. The first step in solving for x often involves applying inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. If x is being added to a number, we can subtract that number from both sides of the equation to isolate x. Similarly, if x is being multiplied by a number, we can divide both sides of the equation by that number. For instance, if we have the equation x + 5 = 12, we can subtract 5 from both sides to get x = 7. If we have the equation 3x = 15, we can divide both sides by 3 to get x = 5. Combining like terms is another important step in solving for x. As discussed previously, like terms are terms that contain the same variable raised to the same power. Combining like terms simplifies the equation and makes it easier to isolate x. For example, if we have the equation 2x + 3x - 4 = 11, we can combine 2x and 3x to get 5x - 4 = 11. Next, we can add 4 to both sides to get 5x = 15, and then divide both sides by 5 to get x = 3. Properties of equality play a crucial role in solving for x. The addition property of equality states that if we add the same number to both sides of an equation, the equality remains true. The subtraction property of equality states that if we subtract the same number from both sides of an equation, the equality remains true. The multiplication property of equality states that if we multiply both sides of an equation by the same number, the equality remains true. The division property of equality states that if we divide both sides of an equation by the same non-zero number, the equality remains true. These properties allow us to manipulate equations while maintaining their balance. In more complex equations, solving for x may involve multiple steps and the application of several algebraic techniques. For instance, we may need to apply the distributive property, clear fractions, or factor polynomials before we can isolate x. The key is to work methodically and apply the appropriate operations in the correct order. Once we have isolated x on one side of the equation, we have found the solution. However, it is always a good idea to check the solution by substituting it back into the original equation to ensure that it satisfies the equation. This helps to catch any errors that may have been made during the solving process.
6. Checking the Solution
Substitute x = 8 into the original equation:
rac{2}{8}+2= rac{2}{3 * 8}+ rac{13}{6}
rac{1}{4}+2= rac{1}{12}+ rac{13}{6}
rac{9}{4}= rac{27}{12}
rac{9}{4}= rac{9}{4}
The solution is correct. Checking the solution is a critical step in the process of solving equations, as it verifies whether the value(s) obtained for the variable(s) actually satisfy the original equation. This step helps to ensure the accuracy of the solution and to identify any errors that may have been made during the solving process. The basic idea behind checking the solution is to substitute the value(s) obtained for the variable(s) back into the original equation. If the substitution results in a true statement, then the solution is correct. If the substitution results in a false statement, then the solution is incorrect, and we need to re-examine our steps to find the mistake. The process of checking the solution typically involves the following steps: First, identify the original equation that was solved. This is important because we want to verify that the solution satisfies the original problem statement. Second, substitute the value(s) obtained for the variable(s) into the original equation. This means replacing every instance of the variable(s) with the value(s) that were found. Third, simplify both sides of the equation using the order of operations (PEMDAS/BODMAS). This involves performing any arithmetic operations, such as addition, subtraction, multiplication, and division, in the correct order. Fourth, compare the simplified expressions on both sides of the equation. If the expressions are equal, then the solution is correct. If the expressions are not equal, then the solution is incorrect. For example, suppose we have solved the equation 2x + 3 = 7 and obtained the solution x = 2. To check this solution, we substitute x = 2 into the original equation: 2(2) + 3 = 7. Simplifying the left side gives 4 + 3 = 7, which is a true statement. Therefore, the solution x = 2 is correct. As another example, suppose we have solved the equation $x^2 - 4 = 0$ and obtained the solutions x = 2 and x = -2. To check these solutions, we substitute each value back into the original equation. For x = 2, we have $2^2 - 4 = 0$, which simplifies to 4 - 4 = 0, a true statement. For x = -2, we have $(-2)^2 - 4 = 0$, which simplifies to 4 - 4 = 0, also a true statement. Therefore, both solutions x = 2 and x = -2 are correct. In some cases, checking the solution may reveal extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but do not satisfy the original equation. This often occurs when solving rational equations or radical equations, where certain operations can introduce solutions that are not valid in the original equation. By checking the solution, we can identify and discard any extraneous solutions. In summary, checking the solution is an essential step in the problem-solving process. It provides a way to verify the accuracy of the solution and to identify any errors or extraneous solutions. By substituting the value(s) obtained for the variable(s) back into the original equation and simplifying, we can determine whether the solution is correct and gain confidence in our answer.
Conclusion
By following these steps, we have successfully solved the equation $ rac{2}{x}+2= rac{2}{3 x}+ rac{13}{6}$, and found the solution to be x = 8. This detailed guide provides a comprehensive understanding of the process, ensuring you can confidently tackle similar equations in the future. Remember to always check your solution to ensure accuracy. The journey through solving equations is a testament to the power and beauty of mathematics. It's a process that not only sharpens our analytical skills but also enhances our problem-solving capabilities. Each equation presents a unique challenge, a puzzle to be unraveled, and the satisfaction of finding the solution is akin to completing a complex jigsaw. The equation $ rac{2}{x}+2= rac{2}{3 x}+ rac{13}{6}$, which we've dissected in this article, serves as an excellent example of how seemingly intricate problems can be systematically approached and resolved. We began by understanding the equation, recognizing it as a rational equation with fractions and variables in the denominator. This initial assessment is crucial, as it sets the stage for the appropriate strategies and techniques to be employed. Next, we identified the least common denominator (LCD), a pivotal step in clearing the fractions. The LCD acts as a common multiple that allows us to transform the equation into a more manageable form. Multiplying both sides of the equation by the LCD was the next logical step, effectively eliminating the fractions and paving the way for simplification. Simplification is where we combined like terms and streamlined the equation, making it easier to isolate the variable x. This stage often involves applying various algebraic properties and operations, such as the distributive property and the order of operations. Solving for x is the heart of the process, where we meticulously applied inverse operations to peel away the layers surrounding x, gradually revealing its value. This requires a keen eye for detail and a solid understanding of algebraic principles. Finally, we arrived at the solution, x = 8. However, our journey didn't end there. We emphasized the importance of checking the solution, a step that is often overlooked but is crucial for ensuring accuracy. By substituting x = 8 back into the original equation, we verified that it indeed satisfied the equation, giving us confidence in our answer. In conclusion, solving equations is not just about finding the right answer; it's about the journey of exploration and discovery. It's about understanding the underlying principles, applying logical reasoning, and persevering through challenges. The equation $ rac{2}{x}+2= rac{2}{3 x}+ rac{13}{6}$ has served as a valuable case study, illustrating the step-by-step process of solving rational equations. With practice and a solid foundation in algebraic techniques, you can confidently tackle any equation that comes your way. Mathematics is a language, and equations are its sentences. By learning to solve equations, we become fluent in this language, capable of expressing and solving a wide range of problems.
