Solving (a+1)/a = 3 + (a-4)/(3a) A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of solving the algebraic equation (a+1)/a = 3 + (a-4)/(3a). This type of equation, involving fractions and variables, is a fundamental concept in algebra, and mastering its solution is crucial for advancing in mathematics. We will break down the steps involved, providing a clear and detailed explanation for each, ensuring a thorough understanding of the process. Understanding the intricacies of solving such equations not only enhances your algebraic skills but also lays a strong foundation for tackling more complex mathematical problems. This guide aims to provide a step-by-step approach, making it accessible to learners of all levels.

To begin, our primary goal is to isolate the variable 'a' on one side of the equation. This involves a series of algebraic manipulations, each designed to simplify the equation while maintaining its balance. The first crucial step is to eliminate the fractions, as they often complicate the process. We achieve this by finding the least common multiple (LCM) of the denominators and multiplying both sides of the equation by it. In this case, the denominators are 'a' and '3a', and their LCM is '3a'. Multiplying both sides by '3a' will effectively clear the fractions, making the equation easier to work with. This step is pivotal, as it transforms a complex fractional equation into a simpler linear one, paving the way for further simplification and solution.

After clearing the fractions, the equation will be in a more manageable form. The next step involves simplifying both sides of the equation by distributing any multiplication and combining like terms. This process of simplification is essential to reduce the equation to its most basic form, making it easier to identify the next steps towards isolating the variable. Careful attention to detail is crucial in this stage, as any error in distribution or combining like terms can lead to an incorrect solution. Once simplified, the equation will likely involve terms with 'a' and constant terms. The next task is to gather all terms involving 'a' on one side of the equation and all constant terms on the other side. This is achieved by adding or subtracting terms from both sides, always ensuring that the equation remains balanced. This rearrangement is a key step in isolating 'a', bringing us closer to the final solution. By systematically moving terms, we create a situation where 'a' is the only variable on one side, making its value readily determinable.

Finally, with all 'a' terms on one side and constants on the other, the last step is to solve for 'a'. This typically involves dividing both sides of the equation by the coefficient of 'a'. This final step directly reveals the value of 'a' that satisfies the original equation. However, it is important to remember that in some cases, there might be restrictions on the value of 'a'. For instance, if the original equation had 'a' in the denominator, we must ensure that our solution does not make the denominator zero, as division by zero is undefined. Therefore, it's crucial to check the solution against the original equation to ensure its validity. This involves substituting the obtained value of 'a' back into the original equation and verifying that both sides are equal. If the solution makes any denominator zero, it is an extraneous solution and must be discarded. This thorough checking process guarantees the accuracy and validity of the solution.

Step-by-step solution

Let's solve the equation step by step:

a+1a=3+a−43a\frac{a+1}{a}=3+\frac{a-4}{3 a}

  1. Eliminate the fractions: Multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 3a.

    3a∗a+1a=3a∗(3+a−43a)3a * \frac{a+1}{a} = 3a * (3 + \frac{a-4}{3a})

    This simplifies to:

    3(a+1)=3a∗3+(a−4)3(a+1) = 3a * 3 + (a-4)

  2. Simplify both sides: Distribute and combine like terms.

    3a+3=9a+a−43a + 3 = 9a + a - 4

    3a+3=10a−43a + 3 = 10a - 4

  3. Rearrange the equation: Move terms involving 'a' to one side and constants to the other side.

    Subtract 3a from both sides:

    3=7a−43 = 7a - 4

    Add 4 to both sides:

    7=7a7 = 7a

  4. Solve for 'a': Divide both sides by the coefficient of 'a', which is 7.

    a=77a = \frac{7}{7}

    a=1a = 1

  5. Check the solution: Substitute a = 1 back into the original equation to ensure it's valid.

    1+11=3+1−43∗1\frac{1+1}{1} = 3 + \frac{1-4}{3 * 1}

    21=3+−33\frac{2}{1} = 3 + \frac{-3}{3}

    2=3−12 = 3 - 1

    2=22 = 2

The solution is valid.

Solution Set

The solution set for the equation is {1}.

Common Mistakes and How to Avoid Them

When solving algebraic equations, it's common to encounter pitfalls that can lead to incorrect solutions. Being aware of these common mistakes and learning how to avoid them is crucial for achieving accuracy and building confidence in your mathematical abilities. One frequent error is the incorrect application of the distributive property. This typically occurs when multiplying a term by an expression enclosed in parentheses. For instance, in the equation we solved, failing to correctly distribute the '3' in the expression '3(a+1)' would lead to an incorrect simplification. To avoid this, always ensure that each term inside the parentheses is multiplied by the term outside. A systematic approach, such as writing out each multiplication step, can help prevent errors. Another common mistake arises when dealing with negative signs. Negative signs can be easily overlooked or mishandled, especially when distributing or combining like terms. Paying close attention to the signs and double-checking each step is essential to prevent errors. A helpful strategy is to rewrite subtraction as addition of a negative number, which can make the manipulation of signs more straightforward.

Another significant source of error is failing to maintain the balance of the equation. Remember, any operation performed on one side of the equation must also be performed on the other side to preserve equality. This principle is fundamental to solving equations, and neglecting it can lead to incorrect solutions. For example, when adding or subtracting terms to isolate the variable, it's crucial to perform the same operation on both sides. A visual aid, such as drawing a line down the middle of the equation to separate the two sides, can help reinforce this concept. Furthermore, errors can arise when combining like terms. This involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. A common mistake is to combine terms that are not like terms, such as adding a term with 'a' to a constant term. To avoid this, carefully identify like terms and group them together before performing any operations. Using different colors or shapes to highlight like terms can be a helpful visual strategy. Lastly, forgetting to check the solution is a critical mistake. Always substitute the obtained solution back into the original equation to verify its validity. This step is crucial for identifying extraneous solutions, which can occur in equations involving fractions or radicals. By checking the solution, you ensure that it satisfies the original equation and that no mathematical errors were made during the solving process.

Tips for Mastering Equation Solving

Mastering equation solving is a fundamental skill in mathematics, and with the right approach and consistent practice, anyone can excel in this area. To begin, a solid understanding of basic algebraic principles is essential. This includes concepts such as the order of operations, the distributive property, and combining like terms. Reviewing these foundational concepts regularly will provide a strong base for tackling more complex equations. Practice is the cornerstone of mastering any mathematical skill, and equation solving is no exception. Start with simpler equations and gradually progress to more challenging ones. This gradual approach allows you to build confidence and develop problem-solving strategies. Regular practice not only reinforces the steps involved in solving equations but also helps you identify patterns and develop intuition. Work through a variety of problems, including those with fractions, decimals, and parentheses, to broaden your skill set.

Another effective strategy is to break down complex equations into smaller, more manageable steps. This involves identifying the key operations that need to be performed and then executing them systematically. Breaking down the problem reduces the cognitive load and makes the solution process less daunting. Writing out each step clearly and neatly is also crucial. This helps to avoid errors and makes it easier to track your progress. Use a systematic approach, such as following a consistent set of steps for each type of equation. For instance, when solving linear equations, a common approach is to first eliminate fractions, then simplify both sides, isolate the variable term, and finally solve for the variable. Developing a systematic approach provides structure and reduces the likelihood of overlooking important steps. Seeking help when needed is also essential. Don't hesitate to ask your teacher, a tutor, or a classmate for assistance if you're struggling with a particular equation or concept. Explaining the problem to someone else can often provide new insights and clarify your understanding. Collaboration and discussion can be powerful tools for learning and problem-solving.

Finally, cultivate a growth mindset towards mathematics. Believe that your mathematical abilities can be developed through dedication and hard work. View mistakes as learning opportunities and don't be discouraged by challenges. Embrace the problem-solving process and focus on understanding the underlying concepts rather than just memorizing formulas or procedures. By adopting a positive and persistent attitude, you can overcome obstacles and achieve mastery in equation solving.