Hey guys! Today, we're diving into the world of algebra to tackle a common problem: solving for y. This is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts. We'll break down the process step by step, using a specific example to illustrate the key techniques. So, grab your pencils and notebooks, and let's get started!
Understanding the Basics of Solving Equations
Before we jump into the specific problem, let's quickly review the basic principles behind solving equations. The main goal is to isolate the variable we're interested in – in this case, 'y' – on one side of the equation. This means we want to manipulate the equation until we have 'y' all by itself on one side, with a numerical value on the other side. Think of it like a mathematical balancing act: whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. This is often referred to as the Golden Rule of Algebra. We can use various operations to achieve this isolation, such as addition, subtraction, multiplication, and division.
When you're solving equations, you are essentially trying to undo the operations that have been performed on the variable. For example, if the equation involves adding a number to 'y', you would subtract that number from both sides to isolate 'y'. Similarly, if 'y' is being multiplied by a number, you would divide both sides by that number. Remember to always perform the same operation on both sides to maintain the balance. Keeping this balance is crucial for arriving at the correct solution. It's like building a house; if the foundation is uneven, the entire structure will be unstable. In algebra, an unbalanced equation leads to a wrong answer. So, always double-check that you're applying the same operation on both sides.
Another key concept is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When simplifying expressions, we follow PEMDAS to ensure we perform the operations in the correct sequence. However, when solving equations, we often work in reverse PEMDAS, undoing the operations in the opposite order. This means we typically deal with addition and subtraction first, then multiplication and division, and so on. This reverse approach helps us systematically peel away the layers of operations that are affecting the variable we want to isolate. By understanding this inverse order of operations, you can effectively strategize your approach to solving various equations.
The Problem: 3y - 5/3 = -5/2 y + 1/2
Okay, let's tackle the specific problem we have: 3y - 5/3 = -5/2 y + 1/2. This equation involves fractions, which might seem intimidating at first, but don't worry! We'll break it down into manageable steps. The first thing we want to do is get all the 'y' terms on one side of the equation and all the constant terms (the numbers without 'y') on the other side. To do this, we'll use addition and subtraction.
The initial equation presents a classic algebraic challenge where we need to isolate the variable 'y'. The presence of fractions might make it appear complex, but with a systematic approach, it becomes quite manageable. The strategy here is to first consolidate all terms involving 'y' on one side of the equation and all constant terms on the other side. This is a standard technique in solving linear equations and forms the foundation for more complex algebraic manipulations. By focusing on this clear objective, we can break down the problem into smaller, more digestible steps. It’s like planning a journey; knowing your destination makes it easier to chart the course.
Before we start moving terms around, it’s helpful to take a moment to observe the equation and identify the different components. We have terms with 'y' on both sides (3y and -5/2 y) and constant terms on both sides (-5/3 and 1/2). This observation helps us plan our next moves strategically. Think of it as surveying the landscape before building a bridge; understanding the terrain helps you choose the best approach. By carefully noting the components of the equation, we can make informed decisions about which operations will most efficiently lead us to the solution. This initial assessment is crucial for optimizing our problem-solving process and minimizing potential errors.
Remember, the core principle in solving any equation is maintaining the balance. Whatever operation we perform on one side, we must perform the same operation on the other side to preserve the equality. This is the fundamental rule that underpins all algebraic manipulations. It's like a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. In the context of our equation, this means that if we add 5/2 y to the left side, we must also add 5/2 y to the right side. This meticulous adherence to balance ensures that the equation remains valid throughout the solving process and that we arrive at the correct solution. Ignoring this principle is akin to building a structure on an unstable foundation; the result will inevitably be flawed.
Step 1: Get the 'y' Terms Together
Let's start by getting all the 'y' terms on the left side of the equation. We have 3y on the left and -5/2 y on the right. To eliminate the -5/2 y term from the right side, we'll add 5/2 y to both sides. This gives us:
3y + 5/2 y - 5/3 = -5/2 y + 5/2 y + 1/2
Notice that -5/2 y + 5/2 y on the right side cancels out, leaving us with:
3y + 5/2 y - 5/3 = 1/2
Now, we need to combine the 'y' terms on the left side. To do this, we need a common denominator for 3 and 5/2. We can rewrite 3y as 6/2 y, so we have:
6/2 y + 5/2 y - 5/3 = 1/2
Combining these terms gives us:
11/2 y - 5/3 = 1/2
The decision to move the 'y' terms to the left side was a strategic one. While it's perfectly valid to move them to the right side instead, our choice here aims to minimize potential negative signs, which can sometimes lead to confusion or errors. Strategic choices like this are an important part of efficient problem-solving in algebra. It's like choosing the most direct route on a map; it saves time and effort. By proactively managing the signs, we can simplify the subsequent steps and reduce the likelihood of making mistakes.
Adding 5/2 y to both sides is a direct application of the fundamental principle of maintaining equality. This operation effectively cancels out the -5/2 y term on the right side, bringing us closer to isolating 'y'. This step exemplifies the concept of inverse operations; we use addition to undo the subtraction. It's like using the opposite action to reverse a previous move. This principle of inverse operations is a cornerstone of algebraic manipulation and is crucial for effectively solving equations.
Combining the 'y' terms, 3y and 5/2 y, requires finding a common denominator. This is a standard procedure when adding or subtracting fractions. The common denominator allows us to express both terms with the same denominator, making it possible to combine their numerators. It's like converting different currencies to a common currency before calculating the total; it ensures we're adding like quantities. Finding the common denominator is a crucial step in simplifying the equation and moving closer to the solution. A misstep here can lead to an incorrect result, so careful attention to detail is essential.
The rewritten equation, 11/2 y - 5/3 = 1/2, represents a significant milestone in our solving process. We have successfully consolidated the 'y' terms on one side, simplifying the equation's structure. This is a crucial step because it brings us closer to isolating 'y' and determining its value. It's like reaching a halfway point in a journey; you've made substantial progress and can see the destination more clearly. This simplified equation now sets the stage for the next steps, where we will focus on isolating 'y' by addressing the constant terms.
Step 2: Get the Constant Terms Together
Next, we want to get all the constant terms on the right side of the equation. We have -5/3 on the left side, so we'll add 5/3 to both sides:
11/2 y - 5/3 + 5/3 = 1/2 + 5/3
This simplifies to:
11/2 y = 1/2 + 5/3
Now, we need to add the fractions on the right side. Again, we need a common denominator, which in this case is 6. So, we rewrite 1/2 as 3/6 and 5/3 as 10/6:
11/2 y = 3/6 + 10/6
Adding these fractions gives us:
11/2 y = 13/6
The decision to move the constant terms to the right side mirrors the strategy we used for the 'y' terms, aiming to group like terms together. This is a fundamental technique in solving equations, as it simplifies the structure and allows us to isolate the variable more effectively. Grouping like terms is like organizing your tools before starting a project; it makes the process smoother and more efficient. By consolidating the constant terms on one side, we create a clearer path towards isolating 'y'.
Adding 5/3 to both sides of the equation is another application of the principle of maintaining equality. This operation effectively cancels out the -5/3 term on the left side, bringing us closer to having 'y' isolated. This step further demonstrates the use of inverse operations, where we use addition to undo subtraction. It's like using an eraser to remove an unwanted mark; it clears the way for what you want to achieve. This meticulous application of inverse operations is crucial for accurately solving equations.
To add the fractions 1/2 and 5/3, we need to find a common denominator, which is 6. This process involves converting each fraction to an equivalent fraction with the common denominator. Finding a common denominator is a necessary step in adding or subtracting fractions, as it ensures we are adding like units. It's like converting measurements to the same scale before adding them; it ensures an accurate result. Paying close attention to this step is essential for avoiding errors in the subsequent calculations.
The equation 11/2 y = 13/6 represents another significant step forward in our solution. We have now isolated the 'y' term on the left side and simplified the constant terms on the right side. This equation is much closer to our final goal of determining the value of 'y'. It's like nearing the end of a maze; you can see the exit. This simplified form sets the stage for the final operation, where we will isolate 'y' completely.
Step 3: Isolate 'y'
Now, we have 11/2 y = 13/6. To isolate 'y', we need to get rid of the 11/2. Since 'y' is being multiplied by 11/2, we'll divide both sides by 11/2. Dividing by a fraction is the same as multiplying by its reciprocal, so we'll multiply both sides by 2/11:
(2/11) * (11/2 y) = (2/11) * (13/6)
On the left side, (2/11) * (11/2) cancels out, leaving us with just 'y':
y = (2/11) * (13/6)
Now, we multiply the fractions on the right side:
y = 26/66
We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:
y = 13/33
Therefore, the solution is y = 13/33.
To finally isolate 'y', we address the coefficient 11/2. Since this coefficient is multiplying 'y', we use the inverse operation: division. Dividing by a fraction is equivalent to multiplying by its reciprocal, which in this case is 2/11. Employing the reciprocal is a common and effective technique in algebra for undoing multiplication by a fraction. It's like using the reverse gear in a car; it takes you back from where you came. This step is crucial for isolating 'y' and determining its value.
Multiplying both sides of the equation by 2/11 maintains the balance, as we are performing the same operation on both sides. This step directly applies the fundamental principle of equality that underpins all algebraic manipulations. It's like keeping the scales balanced when weighing ingredients for a recipe; you need to maintain the proportions. This meticulous adherence to balance ensures that the equation remains valid throughout the solution process.
After multiplying by the reciprocal, we arrive at y = 26/66. This is a valid solution, but it's not in its simplest form. Simplifying fractions is an essential step in algebra, as it presents the solution in the most concise and understandable way. It's like polishing a gem to reveal its brilliance; it enhances the clarity and value. To simplify 26/66, we find the greatest common divisor (GCD) of 26 and 66, which is 2, and divide both the numerator and denominator by it.
The final simplified solution, y = 13/33, represents the culmination of our step-by-step process. We have successfully isolated 'y' and expressed its value as a simplified fraction. This solution is the definitive answer to the original equation. It's like reaching the summit of a mountain after a challenging climb; you've achieved your goal. This final result not only provides the value of 'y' but also demonstrates the power and precision of algebraic techniques.
Final Answer
So, we've successfully solved for y! The final answer is:
y = 13/33
Remember, guys, the key to solving algebraic equations is to break them down into smaller, manageable steps and to always maintain the balance of the equation. Keep practicing, and you'll become a pro at solving for any variable!
Repair Input Keyword
Solve for y in the equation 3y - 5/3 = -5/2 y + 1/2. Simplify your answer as much as possible.
Title
Solve for y Step-by-Step Guide with Examples