Solving For W A Step-by-Step Guide
In the realm of mathematics, solving for a variable is a fundamental skill. It involves isolating the variable on one side of an equation to determine its value. In this article, we will delve into the process of solving for the variable 'w' in the equation w - 4/5 = 2 1/3. We will break down the steps involved, providing a clear and concise explanation to help you grasp the concept. This comprehensive guide will not only walk you through the specific problem but also equip you with the skills to tackle similar algebraic challenges. Understanding how to manipulate equations and isolate variables is crucial for success in various mathematical disciplines, including algebra, calculus, and beyond. So, let's embark on this mathematical journey together and unlock the value of 'w'. Remember, the key to solving equations lies in maintaining balance and performing the same operations on both sides. By mastering these techniques, you'll gain a powerful tool for problem-solving in mathematics and related fields. Let's begin by understanding the basic principles of equation manipulation and then apply them to our specific problem. This step-by-step approach will ensure a clear understanding and build your confidence in tackling algebraic equations.
Understanding the Basics of Equation Manipulation
Before we dive into the specifics of solving for 'w', let's establish a solid foundation by understanding the basic principles of equation manipulation. An equation is a mathematical statement that asserts the equality of two expressions. The goal of solving an equation is to isolate the variable on one side, revealing its value. To achieve this, we employ a set of operations that maintain the balance of the equation. The fundamental principle is that any operation performed on one side of the equation must also be performed on the other side. This ensures that the equality remains valid. These operations include addition, subtraction, multiplication, and division. For example, if we have the equation x + 3 = 5, we can subtract 3 from both sides to isolate 'x'. This gives us x = 2. Similarly, if we have 2y = 10, we can divide both sides by 2 to find the value of 'y', which is 5. Understanding these basic operations is crucial for solving more complex equations. Another important concept is the use of inverse operations. Inverse operations "undo" each other. Addition and subtraction are inverse operations, as are multiplication and division. We use inverse operations to isolate the variable. For instance, to undo addition, we subtract; to undo multiplication, we divide. This principle is at the heart of solving equations. By applying the correct inverse operations, we can systematically isolate the variable and determine its value. Remember, the key is to maintain balance and perform the same operation on both sides of the equation. With a clear understanding of these basic principles, we are well-equipped to tackle the equation w - 4/5 = 2 1/3 and solve for 'w'.
Step 1: Convert the Mixed Number to an Improper Fraction
The first step in solving for 'w' in the equation w - 4/5 = 2 1/3 is to convert the mixed number, 2 1/3, into an improper fraction. A mixed number consists of a whole number and a proper fraction, while an improper fraction has a numerator that is greater than or equal to its denominator. Converting to an improper fraction simplifies the equation and makes it easier to work with. To convert 2 1/3 to an improper fraction, we multiply the whole number (2) by the denominator of the fraction (3) and then add the numerator (1). This result becomes the new numerator, and we keep the same denominator. So, (2 * 3) + 1 = 7. Therefore, 2 1/3 is equivalent to 7/3 as an improper fraction. Now our equation looks like this: w - 4/5 = 7/3. This conversion is a crucial step because it allows us to perform arithmetic operations more easily. Working with fractions often requires a common denominator, and improper fractions make this process smoother. By converting the mixed number to an improper fraction, we have prepared the equation for the next step, which involves isolating 'w'. This step demonstrates the importance of understanding different forms of numbers and how to convert between them. Proficiency in these conversions is essential for success in algebra and other mathematical disciplines. With the mixed number converted, we can now proceed to the next stage of solving for 'w'. Remember, each step we take brings us closer to isolating the variable and finding its value.
Step 2: Isolate 'w' by Adding 4/5 to Both Sides
Now that we have converted the mixed number to an improper fraction, the next step in solving for 'w' is to isolate the variable. In the equation w - 4/5 = 7/3, 'w' is being subtracted by 4/5. To isolate 'w', we need to perform the inverse operation, which is addition. We will add 4/5 to both sides of the equation. This is based on the fundamental principle that any operation performed on one side of the equation must also be performed on the other side to maintain balance. Adding 4/5 to both sides cancels out the -4/5 on the left side, leaving 'w' isolated. The equation now becomes: w - 4/5 + 4/5 = 7/3 + 4/5. Simplifying the left side, we have w = 7/3 + 4/5. This step is crucial because it brings us closer to finding the value of 'w'. By adding 4/5 to both sides, we have successfully isolated 'w' on one side of the equation. The next step involves adding the fractions on the right side of the equation. This requires finding a common denominator, which we will address in the following section. Remember, the goal is to isolate the variable, and by performing the inverse operation, we are making significant progress. This step highlights the importance of understanding inverse operations and their role in solving equations. With 'w' isolated, we can now focus on simplifying the expression on the right side and determining the value of 'w'.
Step 3: Find a Common Denominator and Add the Fractions
Having isolated 'w' in the equation w = 7/3 + 4/5, our next crucial step is to add the fractions on the right side. To do this effectively, we need to find a common denominator for the fractions 7/3 and 4/5. A common denominator is a number that is a multiple of both denominators. In this case, we need to find a common multiple of 3 and 5. The least common multiple (LCM) of 3 and 5 is 15. This will be our common denominator. Now, we need to convert each fraction to an equivalent fraction with a denominator of 15. To convert 7/3 to an equivalent fraction with a denominator of 15, we multiply both the numerator and the denominator by 5 (since 3 * 5 = 15). This gives us (7 * 5) / (3 * 5) = 35/15. Similarly, to convert 4/5 to an equivalent fraction with a denominator of 15, we multiply both the numerator and the denominator by 3 (since 5 * 3 = 15). This gives us (4 * 3) / (5 * 3) = 12/15. Now we can rewrite the equation as w = 35/15 + 12/15. With the fractions having a common denominator, we can now add them by simply adding the numerators and keeping the denominator the same. So, 35/15 + 12/15 = (35 + 12) / 15 = 47/15. Therefore, w = 47/15. This step demonstrates the importance of understanding how to work with fractions, including finding common denominators and converting fractions to equivalent forms. With the fractions added, we have simplified the equation further and are closer to the final solution. The next step will involve expressing the improper fraction as a mixed number, if desired. Remember, mastering fraction manipulation is essential for success in algebra and beyond.
Step 4: Express the Answer as a Mixed Number (Optional)
Having found that w = 47/15, the final step in solving for 'w' is to express the answer as a mixed number. While 47/15 is a perfectly valid answer, it is often preferable to express improper fractions as mixed numbers, especially in certain contexts. A mixed number consists of a whole number and a proper fraction. To convert the improper fraction 47/15 to a mixed number, we need to divide the numerator (47) by the denominator (15). The quotient will be the whole number part of the mixed number, and the remainder will be the numerator of the fractional part. Dividing 47 by 15, we get a quotient of 3 and a remainder of 2. This means that 15 goes into 47 three times with a remainder of 2. Therefore, the whole number part of the mixed number is 3, and the numerator of the fractional part is 2. The denominator remains the same, which is 15. So, 47/15 is equivalent to the mixed number 3 2/15. Therefore, w = 3 2/15. This step is optional, but it demonstrates a complete understanding of fraction manipulation and the ability to express answers in different forms. Converting improper fractions to mixed numbers is a valuable skill in mathematics. By expressing the answer as a mixed number, we have provided a clear and concise solution for 'w'. This final step completes the process of solving for 'w' in the equation w - 4/5 = 2 1/3. Remember, understanding how to convert between improper fractions and mixed numbers is an important aspect of working with fractions. With this skill, you can confidently express your answers in the most appropriate form.
Conclusion: The Value of w
In conclusion, we have successfully solved for 'w' in the equation w - 4/5 = 2 1/3. By following a step-by-step approach, we converted the mixed number to an improper fraction, isolated 'w' by adding 4/5 to both sides, found a common denominator to add the fractions, and finally, expressed the answer as a mixed number. Our journey began with the initial equation and culminated in the solution: w = 3 2/15. This process highlights the importance of understanding fundamental algebraic principles, such as equation manipulation, inverse operations, and fraction arithmetic. Each step we took was crucial in isolating the variable and determining its value. The ability to solve for variables is a cornerstone of mathematics and is essential for success in various fields, including science, engineering, and finance. By mastering these techniques, you gain a powerful tool for problem-solving and critical thinking. Remember, the key to solving equations lies in maintaining balance and performing the same operations on both sides. With practice and a clear understanding of the underlying principles, you can confidently tackle a wide range of algebraic challenges. The value of 'w' in this equation is not just a numerical answer; it represents the culmination of a logical and systematic process. This process is what empowers us to solve problems and make informed decisions in mathematics and beyond. So, embrace the challenge of solving equations, and you will unlock a world of mathematical possibilities.
