Solving For 'y': Leah's Next Equation Step Explained

Hey everyone, let's dive into Leah's math problem! We're helping her figure out the next step in solving an equation for 'y'. Equations like these are super important in algebra, and understanding how to manipulate them is key to mastering the subject. We'll break down the process step-by-step, making it easy to follow along. So, grab your pencils and let's get started. Remember, the goal is to isolate 'y' on one side of the equation. This involves a series of algebraic manipulations to simplify the equation until we get something like 'y = ...' Let's start with the basics of what's going on and then examine Leah's work.

The Foundation: Understanding the Goal

The primary goal in solving for a variable (in this case, 'y') is to isolate it on one side of the equation. Think of it like a treasure hunt; you need to find 'y' and reveal its value. This isolation is achieved by performing inverse operations on both sides of the equation. Remember the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side to keep it balanced. This ensures that the equation remains equivalent throughout the process. For instance, if you add something to the left side, you also have to add it to the right side. And conversely, if you subtract something from the left side, you have to subtract from the right side. And if you multiply or divide either side, do the same to the other side. This principle is fundamental to solving equations correctly. The methods used involve addition, subtraction, multiplication, and division, and sometimes a combination of these.

Before getting into Leah's problem, it's helpful to refresh our memory about some fundamental algebraic concepts. Let's make sure we're all on the same page. Linear equations are the equations that can be written in the form ax + by = c, where a, b, and c are constants and x and y are variables. Leah's equation 3x + 4y = 8 fits this form. When we solve these kinds of equations, we're trying to find values for the variables that make the equation true. In Leah's case, we're focusing on solving for 'y', which means we want to find an equivalent equation where 'y' is by itself on one side. Remember that equivalent equations have the same solution set. The steps Leah is taking are designed to create simpler equivalent equations, bringing her closer to the final answer. These steps always maintain the balance of the equation. Each step is carefully designed to use the inverse operations to isolate the variable we're solving for.

The Importance of Inverse Operations

Inverse operations are at the heart of solving equations. They're like mathematical opposites. Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. These are used to undo operations and isolate the variable. Let's consider a simple example: x + 5 = 10. To isolate 'x', we use the inverse operation of addition (subtraction) and subtract 5 from both sides: x + 5 - 5 = 10 - 5, which simplifies to x = 5. Similarly, if we have 2x = 8, we use the inverse operation of multiplication (division) and divide both sides by 2: 2x / 2 = 8 / 2, which simplifies to x = 4. Using these inverse operations is the key to manipulating equations effectively.

Leah's Equation: A Step-by-Step Breakdown

Leah started with the equation 3x + 4y = 8. Her first step was to subtract 3x from both sides. This is a solid move because the goal is to isolate the term with 'y'. When she subtracted 3x from both sides of the original equation, she obtained 3x - 3x + 4y = 8 - 3x, which simplifies to 4y = 8 - 3x. So, what does Leah do next? Let's figure it out.

Now, let's analyze the equation 4y = 8 - 3x. The term 'y' is currently multiplied by 4. To isolate 'y', we need to get rid of that 4. Since the 4 is multiplied by 'y', the inverse operation we must perform is division. We need to divide both sides of the equation by 4. The division will cancel out the 4 on the left side, leaving 'y' alone and revealing its value. So, based on our understanding of how to solve for 'y', we know that Leah's next step is not to add or subtract anything. Instead, she needs to divide both sides by 4.

Analyzing the Options

Let's evaluate the options Leah has. Option A suggests adding 4y to both sides. Doing this would make the equation more complicated and move us further from isolating 'y'. Option B suggests subtracting 4y from both sides. Subtracting 4y also complicates the equation and does not bring 'y' any closer to being isolated. These steps don't align with the principle of isolating the variable. Option C proposes multiplying both sides of the equation by 1/4. This is the correct approach! Multiplying by 1/4 is the same as dividing by 4, which is the inverse operation that will isolate 'y'. Thus, option C is correct, as dividing both sides of the equation by 4 effectively isolates 'y'.

The Correct Next Step

So, the correct answer is C: Multiply both sides of the equation by 1/4. This will lead to the equation y = (8 - 3x) / 4, and 'y' will be all alone on one side, which is precisely what Leah is trying to achieve. Congratulations, Leah! You are almost there.

Why the Other Options Are Incorrect

Let's clarify why the other options are wrong. Adding or subtracting terms involving 'y' isn't productive at this stage. We want to undo the multiplication by 4, not introduce more 'y' terms. These incorrect steps would not lead us to isolate 'y'. We always need to use inverse operations to get the variable alone.

Conclusion: Leah's Path to 'y'

So, to recap, the key to solving for 'y' is to isolate it by performing the inverse operations of whatever is happening to it in the equation. In Leah's case, after her initial steps, 'y' was multiplied by 4. To undo this, she needs to divide both sides by 4 (or multiply by 1/4). This will leave 'y' all by itself on one side, and Leah will have solved for 'y'. Good job, Leah!

In summary:

• Goal: Isolate 'y' on one side of the equation.
• Method: Use inverse operations (addition/subtraction, multiplication/division) on both sides.
• Leah's Next Step: Multiply both sides of the equation by 1/4.

By following these steps, anyone can master solving equations like Leah, and you will become experts at it too! Keep practicing, and always remember to keep the equation balanced.