Solving Y - 1/2 = -1/7 A Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. It's a cornerstone of algebra and essential for tackling more complex problems in various fields, from physics and engineering to economics and computer science. At its heart, solving an equation means finding the value(s) of the unknown variable(s) that make the equation true. Linear equations, in particular, are the simplest type of equations, but mastering them is crucial for building a strong mathematical foundation. This article will delve into a step-by-step approach to solving a specific linear equation: y - 1/2 = -1/7. We will break down the problem, explain the underlying principles, and provide clear, concise steps to arrive at the solution. Understanding the mechanics behind solving this equation will equip you with the tools to tackle similar problems with confidence.

The equation y - 1/2 = -1/7 is a linear equation because the variable y is raised to the power of 1. To solve for y, we need to isolate it on one side of the equation. This involves performing algebraic manipulations, ensuring that we maintain the balance of the equation. The golden rule of equation solving is: whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This ensures that the equality remains valid throughout the process. In our case, we need to get rid of the -1/2 term on the left side of the equation. To do this, we will add 1/2 to both sides. This is based on the additive property of equality, which states that adding the same number to both sides of an equation does not change the solution. By adding 1/2 to both sides, we effectively cancel out the -1/2 term on the left, leaving y isolated. The next step involves performing the arithmetic on the right side of the equation. We need to add -1/7 and 1/2. This requires finding a common denominator for the fractions. The least common multiple of 7 and 2 is 14, so we will convert both fractions to have a denominator of 14. Once we have a common denominator, we can add the numerators. The resulting fraction will be the value of y. Finally, we will simplify the fraction to its lowest terms, if necessary. This gives us the solution for y. We can then check our solution by substituting it back into the original equation. If the equation holds true, we have found the correct solution. This process of solving equations is not just about finding the answer; it's about understanding the underlying logic and developing a systematic approach to problem-solving.

To effectively solve the equation y - 1/2 = -1/7, we will undertake a meticulous, step-by-step process. This approach not only yields the correct answer but also reinforces the fundamental principles of algebraic manipulation. Our primary objective is to isolate the variable y on one side of the equation. This is achieved by performing inverse operations, ensuring the equation's balance throughout. Let's begin by meticulously detailing each step involved in solving this equation.

  1. Isolate the Variable: The initial step is to isolate the variable y. To accomplish this, we need to eliminate the term -1/2 from the left side of the equation. This is achieved by adding its additive inverse, +1/2, to both sides of the equation. This operation is grounded in the fundamental principle of maintaining equality; whatever we do to one side, we must replicate on the other. This ensures that the equation remains balanced and the solution remains valid. The equation now looks like this:

    y - 1/2 + 1/2 = -1/7 + 1/2

  2. Simplify the Left Side: On the left side of the equation, -1/2 and +1/2 cancel each other out, leaving us with just y. This simplification is a direct consequence of the additive inverse property, where a number added to its inverse equals zero. This step effectively isolates the variable y on the left side, bringing us closer to the solution.

    y = -1/7 + 1/2

  3. Find a Common Denominator: Now, we focus on the right side of the equation, where we need to add two fractions: -1/7 and 1/2. Before we can add fractions, they must have a common denominator. The least common multiple (LCM) of 7 and 2 is 14. This means we need to convert both fractions to equivalent fractions with a denominator of 14. To do this, we multiply the numerator and denominator of -1/7 by 2, resulting in -2/14. Similarly, we multiply the numerator and denominator of 1/2 by 7, resulting in 7/14. Now, the equation becomes:

    y = -2/14 + 7/14

  4. Add the Fractions: With a common denominator of 14, we can now add the fractions on the right side. This involves adding the numerators while keeping the denominator the same. So, -2 + 7 equals 5. Therefore, the sum of the fractions is 5/14.

    y = 5/14

  5. The Solution: After performing the necessary algebraic manipulations and arithmetic operations, we arrive at the solution: y = 5/14. This value represents the solution to the original equation, meaning that if we substitute 5/14 for y in the original equation, the equation will hold true. This is a crucial step in the problem-solving process.

  6. Verification (Optional but Recommended): To ensure the accuracy of our solution, we can perform a verification step. This involves substituting the value of y we found (5/14) back into the original equation and checking if both sides of the equation are equal. If they are, our solution is correct. Let's substitute y = 5/14 into the original equation:

    5/14 - 1/2 = -1/7

    To verify, we need to find a common denominator for 14 and 2, which is 14. So, we rewrite 1/2 as 7/14:

    5/14 - 7/14 = -1/7

    Now, subtract the fractions on the left side:

    (5 - 7)/14 = -2/14

    Simplify the fraction:

    -2/14 = -1/7

    The left side now equals -1/7, which is the same as the right side of the original equation. This confirms that our solution, y = 5/14, is indeed correct. This verification step adds a layer of confidence to our solution and reinforces the understanding of equation solving.

Solving equations can sometimes be tricky, and even experienced mathematicians can make mistakes. However, by being aware of common pitfalls, you can significantly improve your accuracy and problem-solving skills. This section will highlight some of the most frequent errors students make when solving equations like y - 1/2 = -1/7 and provide strategies to avoid them. Recognizing these potential stumbling blocks can make the equation-solving process smoother and more efficient.

  1. Incorrectly Applying the Order of Operations: One of the most common mistakes is not following the order of operations (PEMDAS/BODMAS). While this equation is relatively simple, it's crucial to understand the principle for more complex problems. In our case, we are dealing with addition and subtraction, which are performed from left to right. However, in other equations, you might encounter parentheses, exponents, multiplication, and division, which must be addressed in the correct order. Forgetting this order can lead to incorrect results. For example, if there were a multiplication operation before the subtraction, it should be performed first. To avoid this, always write out each step clearly and double-check that you're applying the correct operation at each stage. This methodical approach can significantly reduce errors.

  2. Errors with Fraction Arithmetic: Working with fractions can be challenging for many students. Common mistakes include forgetting to find a common denominator before adding or subtracting fractions, incorrectly multiplying or dividing fractions, or simplifying fractions improperly. In our equation, we needed to add -1/7 and 1/2, which required finding a common denominator of 14. If you were to add the fractions without a common denominator, you would arrive at an incorrect answer. Another mistake is incorrectly converting fractions to equivalent fractions with the common denominator. For instance, multiplying only the numerator or only the denominator will result in an incorrect fraction. To avoid these errors, always double-check your fraction arithmetic. Write out each step of the fraction operations clearly, and use visual aids like fraction bars or diagrams if they help you understand the concepts better. Practice with a variety of fraction problems can also build your confidence and accuracy.

  3. Sign Errors: Sign errors are another common source of mistakes. This often happens when adding or subtracting negative numbers. In our equation, we added 1/2 to both sides to isolate y. It's crucial to correctly handle the negative sign on -1/7. A simple sign error can completely change the result. For example, if you incorrectly added 1/2 to -1/7 and got a positive result instead of a smaller negative or a positive fraction, your final answer would be wrong. To minimize sign errors, pay close attention to the signs of the numbers and use the rules of sign arithmetic carefully. It can be helpful to rewrite the equation with the signs explicitly stated, such as y + (-1/2) = -1/7. This can make it easier to keep track of the signs. Also, consider using a number line to visualize addition and subtraction of signed numbers, which can help you understand the concept better.

  4. Not Performing the Same Operation on Both Sides: The fundamental principle of solving equations is to maintain balance by performing the same operation on both sides. Forgetting to do this is a major error that will lead to an incorrect solution. In our equation, we added 1/2 to both sides to isolate y. If we had only added 1/2 to the left side, the equation would no longer be balanced, and the solution would be wrong. To avoid this mistake, always make a conscious effort to perform the same operation on both sides of the equation. You can even draw a vertical line down the equals sign to remind yourself to treat both sides equally. Write out each step clearly, showing the operation you're performing on both sides. This will help you stay organized and prevent you from forgetting to balance the equation.

  5. Skipping Steps: While it might be tempting to skip steps to save time, this can often lead to errors. Each step in solving an equation is crucial, and skipping a step increases the likelihood of making a mistake. For example, skipping the step of finding a common denominator when adding fractions can lead to an incorrect result. Similarly, skipping the step of writing out the addition or subtraction operation can make it easier to make a sign error. To avoid skipping steps, adopt a methodical approach to problem-solving. Write out each step clearly and concisely, showing all the necessary operations. This will help you keep track of your work and reduce the chances of making mistakes. It might take a little more time initially, but it will ultimately save you time and frustration by ensuring accuracy.

Mathematics and equation solving aren't just abstract concepts confined to textbooks and classrooms; they have a plethora of real-world applications that touch our lives every day. Linear equations, in particular, form the bedrock for numerous practical calculations and problem-solving scenarios. Understanding how to solve them not only enhances your mathematical skills but also equips you with valuable tools for navigating various aspects of life and work. This section explores some compelling real-world applications of solving linear equations, highlighting their relevance and importance.

  1. Budgeting and Personal Finance: Linear equations are instrumental in managing personal finances and creating budgets. For instance, consider the equation: Total Expenses = Fixed Expenses + (Variable Expense per Unit × Number of Units). If you know your total expenses, fixed expenses, and variable expense per unit, you can use a linear equation to determine the number of units you can afford. Similarly, you can use linear equations to calculate loan payments, savings goals, and investment returns. By setting up an equation that represents your financial situation, you can solve for unknown variables, such as the amount you need to save each month to reach a specific goal or the number of years it will take to pay off a loan. This application of linear equations empowers individuals to make informed financial decisions and plan for the future.

  2. Cooking and Baking: Cooking and baking often involve scaling recipes up or down, which requires solving linear equations. For example, if a recipe calls for 2 cups of flour and yields 12 cookies, you can use a linear equation to determine how much flour you need to make 30 cookies. The equation would be: (Flour Needed / 30 Cookies) = (2 Cups / 12 Cookies). Solving for