Solving 4y + 9 = 7y + 6 Step-by-Step Guide With Solution

Hey guys! Today, we're diving into a common type of algebra problem: solving linear equations. Specifically, we're going to tackle the equation 4y + 9 = 7y + 6. This kind of problem might seem tricky at first, but with a few simple steps, you'll be solving these like a pro. We'll break down each step, making sure you understand the logic behind it, not just the mechanics. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what this equation is telling us. The equation 4y + 9 = 7y + 6 is a statement that two expressions, 4y + 9 and 7y + 6, are equal. Our goal is to find the value of the variable y that makes this statement true. Think of it like a puzzle – we need to find the piece (the value of y) that fits perfectly.

The variable y represents an unknown number. The equation tells us that if we multiply this number by 4 and add 9, we'll get the same result as if we multiply it by 7 and add 6. This might seem a bit abstract, but that's the beauty of algebra – it allows us to work with unknowns in a precise way.

To solve for y, we need to isolate it on one side of the equation. This means we want to get y all by itself on either the left-hand side (LHS) or the right-hand side (RHS). We can do this by performing operations on both sides of the equation. The key is to maintain balance – whatever we do to one side, we must also do to the other. This ensures that the equality remains true.

Isolating the Variable: The Core Strategy

The core strategy for solving linear equations like 4y + 9 = 7y + 6 involves isolating the variable. This means getting the term with 'y' on one side of the equation and the constant terms (the numbers without 'y') on the other side. To do this, we'll use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations.

Our goal is to manipulate the equation by adding, subtracting, multiplying, or dividing both sides by the same value, always with the aim of simplifying the equation and bringing us closer to isolating 'y'. Think of it like peeling away layers of an onion – we're gradually stripping away the extra terms until we're left with just 'y' on one side.

The process may seem a bit like a dance, moving terms from one side to the other while maintaining the balance of the equation. But with each step, we're making progress towards our ultimate goal: finding the value of 'y' that satisfies the equation. So, let's dive into the steps and see how this works in practice.

Step-by-Step Solution

Now, let's walk through the steps to solve the equation 4y + 9 = 7y + 6. We'll break it down into manageable chunks, explaining the reasoning behind each step.

Step 1: Grouping the 'y' Terms

Our first goal is to get all the terms with y on the same side of the equation. Looking at the equation 4y + 9 = 7y + 6, we have 4y on the left side and 7y on the right side. A common strategy is to move the smaller y term to the side with the larger y term. This helps to avoid negative coefficients, which can sometimes be tricky to work with.

In this case, 4y is smaller than 7y, so we'll move the 4y term to the right side. To do this, we subtract 4y from both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain balance.

So, we have:

4y + 9 - 4y = 7y + 6 - 4y

On the left side, 4y and -4y cancel each other out, leaving us with just 9. On the right side, 7y - 4y simplifies to 3y. Our equation now looks like this:

9 = 3y + 6

We've made progress! We've successfully grouped the y terms on one side of the equation.

Step 2: Grouping the Constant Terms

Next, we want to group the constant terms (the numbers without y) on the other side of the equation. In our current equation, 9 = 3y + 6, we have the constant term 9 on the left side and the constant term 6 on the right side. We want to move the 6 to the left side so that all the constant terms are together.

To do this, we subtract 6 from both sides of the equation. Again, we're using the inverse operation (subtraction) to move the term to the other side.

9 - 6 = 3y + 6 - 6

On the left side, 9 - 6 simplifies to 3. On the right side, 6 - 6 cancels out, leaving us with just 3y. Our equation now looks like this:

3 = 3y

We're getting closer! We've now grouped the y terms on one side and the constant terms on the other side.

Step 3: Isolating 'y'

Finally, we need to isolate y completely. In our equation, 3 = 3y, the y is being multiplied by 3. To undo this multiplication, we use the inverse operation: division. We divide both sides of the equation by 3.

3 / 3 = 3y / 3

On the left side, 3 / 3 simplifies to 1. On the right side, 3y / 3 simplifies to y. Our equation now looks like this:

1 = y

Or, more commonly written:

y = 1

We've done it! We've isolated y and found its value. The solution to the equation 4y + 9 = 7y + 6 is y = 1.

Checking the Solution

It's always a good idea to check your solution to make sure you haven't made any mistakes. To check our solution, we substitute the value we found for y (which is 1) back into the original equation, 4y + 9 = 7y + 6, and see if both sides are equal.

Substituting y = 1

Let's substitute y = 1 into the left-hand side (LHS) of the equation:

4y + 9 = 4(1) + 9 = 4 + 9 = 13

Now, let's substitute y = 1 into the right-hand side (RHS) of the equation:

7y + 6 = 7(1) + 6 = 7 + 6 = 13

We see that both the LHS and the RHS equal 13 when y = 1. This confirms that our solution is correct. Phew!

Why Checking is Crucial

Checking your solution is a crucial step in solving equations. It's like a safety net that catches any errors you might have made along the way. By substituting your solution back into the original equation, you can quickly verify whether your answer is correct.

Sometimes, errors can creep in during the algebraic manipulations – a missed sign, an incorrect operation, or a simple arithmetic mistake. Checking your solution helps you catch these errors before they become a bigger problem. It gives you confidence in your answer and ensures that you're on the right track.

Evaluating the Given Options

The original question presented us with two possible solutions: y = -3 and y = 1. We've already determined that y = 1 is the correct solution. But let's also see why y = -3 is not a solution. This will help us understand how to evaluate potential solutions in general.

Testing y = -3

Let's substitute y = -3 into the original equation, 4y + 9 = 7y + 6, and see what happens.

First, we substitute y = -3 into the left-hand side (LHS):

4y + 9 = 4(-3) + 9 = -12 + 9 = -3

Now, let's substitute y = -3 into the right-hand side (RHS):

7y + 6 = 7(-3) + 6 = -21 + 6 = -15

We see that when y = -3, the LHS is -3 and the RHS is -15. Since -3 is not equal to -15, y = -3 is not a solution to the equation.

The Importance of Verification

This exercise highlights the importance of verifying potential solutions. Even if a value seems like it might be a solution, it's crucial to substitute it back into the original equation to confirm. This is the definitive way to determine whether a value satisfies the equation.

Conclusion: Mastering Linear Equations

So, guys, we've successfully solved the equation 4y + 9 = 7y + 6 and found that the solution is y = 1. We also saw why y = -3 is not a solution. We did this by carefully following the steps of isolating the variable and checking our answer.

The key takeaways from this exercise are:

1. Isolate the variable: Get the y terms on one side and the constant terms on the other.
2. Use inverse operations: Add or subtract to move terms across the equals sign, and multiply or divide to isolate y.
3. Check your solution: Substitute your answer back into the original equation to verify that it works.

Solving linear equations is a fundamental skill in algebra, and it's something you'll use again and again in more advanced math courses. By mastering these basic steps, you'll build a solid foundation for future success. Keep practicing, and you'll become a whiz at solving equations in no time!

If you guys have any questions or want to try some more examples, feel free to ask! Keep up the great work!