Solving For Y: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for y. We'll tackle the equation 3(y+2) + y = 4(y-1) + 9 step-by-step, making sure even those new to algebra can follow along. This isn't just about getting the answer; it's about understanding the process. Think of it like learning a new recipe – we'll break down each ingredient and step, so you can confidently solve similar equations in the future. Understanding how to solve for a variable is a fundamental skill in mathematics, popping up everywhere from basic equations to complex scientific formulas. Let's get started and demystify the process of solving this equation. We'll make sure you understand the 'why' behind each step.

First things first: understanding the basics. Our main goal is to isolate y on one side of the equation. This means we need to get y by itself, with a single value on the other side. We do this using a combination of algebraic rules, mainly the distributive property and the properties of equality. Remember, what you do to one side of the equation, you must do to the other side to keep things balanced. It's like a seesaw – if you add weight to one side, you have to add the same weight to the other to keep it level. This principle is key. We'll also use simplification rules, such as combining like terms. Like terms are those with the same variable raised to the same power, or just plain numbers. For instance, 3y and y are like terms because they both have y, while 3y and y^2 are not. Similarly, 2 and 9 are like terms.

Let’s look at the original equation once more, 3(y+2) + y = 4(y-1) + 9. The first step in solving is often to simplify both sides of the equation. This will involve using the distributive property, which says that a(b + c) = ab + ac. For the left side, we need to multiply everything inside the parenthesis by the number outside. Similarly, for the right side. This essentially means that we need to multiply the number outside the parentheses by each term inside the parentheses. After that, we must combine like terms and move all terms containing y to one side and all constants to the other. Finally, we'll isolate y by dividing both sides by the coefficient of y. The distributive property is one of the most important concepts in algebra, so understanding it well can unlock many other topics in math. It simplifies complex equations, and it is a gateway for understanding more advanced algebraic principles. This process makes the equation easier to manage, allowing us to eventually isolate y and find its value. So, are you ready to simplify and solve?

Step-by-Step Solution

Alright, let's break down the process step-by-step to solve for y in the equation 3(y+2) + y = 4(y-1) + 9.

Step 1: Distribute

First, we're going to apply the distributive property. This means multiplying the number outside each set of parentheses by each term inside. On the left side, we have 3(y+2). Multiplying, we get 3*y + 3*2, which simplifies to 3y + 6. Now, on the right side, we have 4(y-1). Multiplying, we get 4*y - 4*1, which simplifies to 4y - 4. So now, the equation looks like this: 3y + 6 + y = 4y - 4 + 9. Remember, applying the distributive property helps to get rid of the parentheses and makes it easier to combine like terms. The most common mistakes are related to the sign, so pay close attention to the positive and negative signs. Always remember, a negative number multiplies with a positive number equals to a negative number, a negative number multiplies by a negative number equals to a positive number, and a positive number multiplies with a positive number equals to a positive number.

Why do we do this? The distributive property is vital for simplifying the equation and getting ready to isolate y. It clears out the parentheses, making the equation more manageable. This also clarifies the individual terms, letting us combine like terms effectively. By distributing, we are essentially making the equation cleaner and more straightforward. It is like organizing a messy desk before starting work, it helps to focus on the essential task at hand, which is to solve for y. This is the first essential step in solving equations like these.

Step 2: Combine Like Terms

Next, we'll simplify both sides of the equation by combining like terms. On the left side, we have 3y + 6 + y. The like terms here are 3y and y. Combining these gives us 4y. So, the left side simplifies to 4y + 6. On the right side, we have 4y - 4 + 9. Here, the like terms are -4 and 9. Combining these gives us 5. So, the right side simplifies to 4y + 5. Now, the equation looks like this: 4y + 6 = 4y + 5. Combining like terms reduces the number of terms we need to manage, making the equation easier to work with. It's like grouping similar objects together to make a process more efficient.

Why combine like terms? Combining like terms is all about streamlining the equation. It makes it shorter and easier to see what we're working with. It's similar to simplifying a recipe by combining similar ingredients. After combining the like terms, it is easier to solve and find the value of y. Also, this ensures that each term is properly accounted for, and this lowers the chance of errors. This also helps to reduce the chance of making a mistake later on during the solving stage.

Step 3: Isolate the Variable

Now, let's work on isolating the variable, y. We want to get all terms with y on one side of the equation and the constants on the other. First, we'll subtract 4y from both sides of the equation 4y + 6 = 4y + 5. Doing so will remove the 4y term from both sides. This gives us 6 = 5. Wait, this seems strange. The y terms have canceled out, and we're left with a statement that isn't true. We're left with an equation where the variable cancels out, which gives us an interesting result.

What happens if we get a false statement? When we arrive at a statement like 6 = 5, it means the equation has no solution. There is no value of y that will make the original equation true. This is a common occurrence in algebra, and it's essential to recognize this outcome.

Why do we isolate the variable? Isolating the variable is the key to finding the solution. It is the core of solving the equation. The goal is to get y all by itself on one side of the equation, making it clear what its value must be to make the equation true. It’s like clearing a path to the answer. By getting the variable alone, we can identify the answer.

Conclusion: No Solution

In conclusion, after meticulously working through the equation 3(y+2) + y = 4(y-1) + 9, we arrived at the statement 6 = 5. This is a false statement, which means that the equation has no solution. This does happen in algebra. It means there is no value of y that can satisfy the original equation. So, the final answer is that there is no solution to this particular equation. This is a good lesson in understanding what results in an equation mean.

Why is this important? Understanding the different types of solutions is an important part of solving equations. It helps you understand the properties of algebra and ensures you understand how equations work. Recognizing when there is no solution, or when there are infinitely many solutions, is a critical skill in algebra. The solution can give you a better grasp of the mathematical concepts. So, keep practicing, and you will become more proficient in solving equations and understanding all of the different types of results you can get!

This is just one example. You can apply these steps to many equations to solve them. Remember to review and practice the concepts to sharpen your skills. With regular practice, solving for y and similar equations will become second nature! Good luck, and happy solving!