Solving Linear Equations Step By Step (5x - 16)/6 = (x + 8)/12 + (x + 1)/3
This article provides a detailed, step-by-step solution to the linear equation . We will explore the fundamental concepts of solving linear equations, emphasizing clarity and precision in each step. Understanding how to solve such equations is crucial in various fields, including mathematics, physics, engineering, and economics. Linear equations, characterized by a variable raised to the power of one, form the backbone of many mathematical models. Mastering the techniques to solve them not only enhances problem-solving skills but also builds a solid foundation for more advanced mathematical concepts. This guide aims to empower readers with the confidence and knowledge to tackle similar problems effectively.
Understanding the Problem
The given equation is a linear equation involving fractions. The goal is to find the value of x
that satisfies the equation. To solve this, we need to eliminate the fractions, simplify the equation, and isolate x
on one side. The process involves finding a common denominator, combining like terms, and applying algebraic manipulations to maintain the equation's balance. A clear understanding of these principles is essential for accurate problem-solving. Linear equations are ubiquitous in mathematics and its applications, appearing in various contexts from simple word problems to complex modeling scenarios. The ability to confidently solve these equations is a foundational skill that opens doors to more advanced mathematical concepts and real-world applications.
Step 1: Find the Least Common Denominator (LCD)
To eliminate the fractions, we first need to find the least common denominator (LCD) of the denominators 6, 12, and 3. The LCD is the smallest number that is a multiple of all the denominators. In this case, the LCD of 6, 12, and 3 is 12. Identifying the LCD is a crucial step as it allows us to transform the equation into a more manageable form without fractions. The concept of LCD is fundamental in arithmetic and algebra, playing a vital role in operations involving fractions, such as addition, subtraction, and equation solving. A thorough understanding of LCD not only simplifies calculations but also provides a deeper insight into the structure of numbers and their relationships. Finding the LCD often involves prime factorization, a technique that breaks down numbers into their prime factors, enabling easier identification of common multiples.
Step 2: Multiply Both Sides of the Equation by the LCD
Next, we multiply both sides of the equation by the LCD, which is 12. This step will clear the fractions. The equation becomes:
Multiplying both sides of an equation by the same non-zero number maintains the equality, a fundamental principle in algebra. This operation eliminates the denominators, transforming the equation into a simpler form that is easier to solve. The distributive property is then applied to expand the multiplication on the right side of the equation, ensuring that each term within the parentheses is multiplied by the LCD. This step is pivotal in simplifying the equation and setting the stage for further algebraic manipulations to isolate the variable. Clearing fractions is a common strategy in solving equations, making the subsequent steps of combining like terms and isolating the variable more straightforward.
Step 3: Simplify the Equation
Now, we simplify both sides of the equation:
Expanding the terms, we get:
Combining like terms on the right side:
Simplifying the equation is a crucial step in the problem-solving process, allowing us to reduce the complexity and make it easier to isolate the variable. This involves applying the distributive property to remove parentheses, and then combining like terms on each side of the equation. The goal is to consolidate terms with the variable and constant terms separately, making the equation more manageable. Accuracy in simplification is paramount, as any error in this step can propagate through the remaining steps and lead to an incorrect solution. This process demonstrates the power of algebraic manipulation in transforming complex expressions into simpler, more understandable forms.
Step 4: Isolate the Variable
To isolate the variable x
, we first subtract 5x
from both sides:
which simplifies to:
Next, add 32 to both sides:
giving us:
Isolating the variable is the key step in solving any equation. It involves performing algebraic operations on both sides of the equation to get the variable term alone on one side. The operations performed must maintain the equality of the equation, ensuring that the solution remains valid. This typically involves adding or subtracting terms to move them across the equality sign, and then multiplying or dividing to eliminate coefficients. The process of isolating the variable demonstrates the power of algebraic manipulation in solving equations and revealing the unknown value. It requires careful attention to detail and a solid understanding of the properties of equality.
Step 5: Solve for x
Finally, divide both sides by 5 to solve for x
:
which gives us:
Step 6: Final Answer
Thus, the solution to the equation is:
The correct answer is not among the options.
Solving for the variable involves isolating it completely on one side of the equation, revealing its value. This is typically the final step in the solution process, where the coefficient of the variable is divided from both sides to obtain the value of the variable. The solution should be carefully checked by substituting it back into the original equation to ensure that it satisfies the equality. Accuracy in this final step is crucial, as it determines the correctness of the entire solution. Solving for the variable demonstrates the culmination of all the algebraic manipulations performed and provides the answer to the problem.
Conclusion
In this article, we have demonstrated a step-by-step solution to the linear equation . By finding the LCD, eliminating fractions, simplifying the equation, isolating the variable, and solving for x
, we arrived at the solution . This process highlights the importance of understanding fundamental algebraic principles in solving mathematical problems. Mastering these techniques not only enhances mathematical proficiency but also develops critical thinking and problem-solving skills applicable in various real-world scenarios. The ability to confidently solve linear equations is a foundational skill that underpins many advanced mathematical concepts and practical applications.