Solving For Y In The Equation 12y + D = -19y + T

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When dealing with algebraic equations, isolating a specific variable is a fundamental skill. In this article, we will be focusing on a common scenario: rewriting an equation to solve for y. The given equation is a linear equation with y as the variable we want to isolate. Understanding how to manipulate equations to solve for a specific variable is crucial in various fields, including mathematics, physics, and engineering. It allows us to express one variable in terms of others, providing valuable insights and enabling us to solve problems efficiently. This step-by-step guide will walk you through the process, providing clear explanations and highlighting key concepts. You'll learn how to apply the principles of algebra to rearrange the equation, ultimately arriving at the correct solution for y. So, let's dive in and master the art of solving for y! The ability to solve for a variable is not just a mathematical exercise; it's a powerful tool for problem-solving in many real-world contexts. Whether you're calculating the trajectory of a projectile, determining the current in an electrical circuit, or optimizing a business model, the ability to manipulate equations and isolate variables is essential. This article will equip you with the skills and understanding you need to confidently tackle these types of problems.

The Problem

Our starting point is the equation:

12y + d = -19y + t

Our goal is to isolate y on one side of the equation. This means we want to rewrite the equation in the form y = (some expression). To achieve this, we'll need to carefully apply algebraic operations, ensuring that we maintain the equality of both sides. The process of solving for a variable involves a series of strategic steps, each designed to bring us closer to our goal. We'll begin by grouping the terms containing y on one side of the equation and the constant terms on the other. This will set the stage for the final step, where we isolate y by dividing both sides of the equation by its coefficient. Throughout the process, we'll emphasize the importance of performing the same operation on both sides of the equation to maintain balance and ensure the validity of our solution. This principle is a cornerstone of algebraic manipulation and is essential for accurate problem-solving.

Step 1: Group the y terms

To begin, we need to get all the terms containing y on the same side of the equation. A common approach is to add 19y to both sides. Adding 19y to both sides of the equation maintains the balance and allows us to consolidate the y terms. This step is crucial because it brings us closer to isolating y. The underlying principle here is the addition property of equality, which states that adding the same quantity to both sides of an equation does not change the equality. By adding 19y, we eliminate the y term from the right side and move it to the left side, where it can be combined with the existing 12y term. This strategic move simplifies the equation and makes it easier to solve for y. Without this step, the y terms would be scattered on both sides, making it difficult to isolate y. The ability to recognize and apply this type of algebraic manipulation is a key skill in solving equations. It demonstrates a deep understanding of the properties of equality and how they can be used to transform equations into more manageable forms.

12y + d + 19y = -19y + t + 19y

This simplifies to:

31y + d = t

Step 2: Isolate the y term

Next, we want to isolate the term with y (which is 31y). To do this, we need to eliminate the constant term d from the left side of the equation. Subtracting d from both sides achieves this. Subtracting d from both sides is another application of the properties of equality. Just as adding the same quantity to both sides maintains balance, so does subtracting the same quantity. By subtracting d, we effectively move it from the left side to the right side, where it can be combined with the other constant term, t. This step is essential for isolating the y term and getting us closer to solving for y. The ability to recognize the need for this type of manipulation is a key aspect of algebraic problem-solving. It demonstrates an understanding of how to use inverse operations to undo unwanted terms and isolate the variable of interest. Without this step, the d term would remain on the left side, preventing us from directly solving for y. The strategic use of subtraction in this way is a powerful tool in algebraic manipulation.

31y + d - d = t - d

This simplifies to:

31y = t - d

Step 3: Solve for y

Finally, to solve for y, we need to get y by itself. Currently, y is being multiplied by 31. To undo this multiplication, we divide both sides of the equation by 31. Dividing both sides by 31 is the final step in isolating y. This step utilizes the division property of equality, which states that dividing both sides of an equation by the same non-zero quantity maintains the equality. By dividing by 31, we effectively undo the multiplication of y by 31, leaving y isolated on the left side. This is the culmination of our efforts, and it provides us with the solution for y in terms of t and d. The ability to perform this type of division is a fundamental skill in algebra. It demonstrates an understanding of how to use inverse operations to solve for variables. Without this step, y would not be fully isolated, and we would not have a solution for its value. The careful and strategic application of division in this way is a key element of algebraic problem-solving.

31y / 31 = (t - d) / 31

This gives us the solution:

y = (t - d) / 31

The Answer

Comparing our solution to the provided options, we see that option A, y = (t - d) / 31, is the correct answer.

Why Other Options Are Incorrect

It's important to understand why the other options are incorrect. This helps solidify your understanding of the process and prevent similar errors in the future.

  • Option B: y = -7(t + d): This option is incorrect because it doesn't follow the correct steps of isolating y. There's no clear algebraic manipulation that would lead to this solution. The multiplication by -7 and the addition of t and d within the parentheses suggest a misunderstanding of the order of operations and the properties of equality.
  • Option C: y = (t + d) / -7: This option is incorrect because it seems to have incorrectly combined terms or applied the division operation. The negative sign in the denominator is also a potential source of error. A careful step-by-step solution, as demonstrated above, will reveal that this option does not follow from the original equation.
  • Option D: y = 31(t - d): This option is incorrect because it multiplies (t - d) by 31 instead of dividing. This is the inverse of the correct operation and indicates a misunderstanding of how to isolate y when it is being multiplied by a coefficient. The correct step, as we saw, is to divide both sides of the equation by 31, not multiply.

By analyzing why these options are incorrect, we reinforce our understanding of the correct solution process and highlight the importance of following each step carefully and accurately. This type of analysis is a valuable learning tool and can help prevent future errors.

Key Takeaways

  • Isolate the variable: The goal is to get the variable you're solving for (in this case, y) by itself on one side of the equation.
  • Use inverse operations: To undo addition, subtract; to undo subtraction, add; to undo multiplication, divide; and to undo division, multiply.
  • Maintain balance: Whatever operation you perform on one side of the equation, you must perform on the other side to keep the equation balanced.
  • Step-by-step approach: Break down the problem into smaller, manageable steps. This makes the process less overwhelming and reduces the chances of errors.

By mastering these key takeaways, you'll be well-equipped to solve a wide range of algebraic equations and confidently tackle problems that require isolating variables. These skills are not just valuable in mathematics but also in various other fields where mathematical modeling and problem-solving are essential. The ability to manipulate equations and isolate variables is a fundamental tool for critical thinking and decision-making.

Practice Problems

To further solidify your understanding, try solving these practice problems:

  1. Solve for x: 5x - 3 = 2x + 9
  2. Solve for a: 4(a + 2) = 16
  3. Solve for z: 7z + 10 = 3z - 6

Working through these problems will give you valuable practice and help you develop your problem-solving skills. Remember to apply the steps and principles we've discussed in this article. If you encounter any difficulties, review the steps and explanations provided earlier. The key to mastering algebra is consistent practice and a willingness to learn from your mistakes. Each problem you solve will build your confidence and strengthen your understanding of the underlying concepts.

By working through this article and the practice problems, you've taken a significant step towards mastering the art of solving for y and other variables in linear equations. Remember to apply these principles to other algebraic problems and continue practicing to hone your skills. The more you practice, the more confident and proficient you'll become in your ability to solve equations and tackle mathematical challenges.

Keywords: solving for y, linear equations, algebraic manipulation, isolate variable, inverse operations, properties of equality