Solving The Equation: 5x + (x + 3) = 38.4 - Math Guide

Hey math enthusiasts! Ever stumbled upon an equation that looks a bit tricky? Don't worry, we've all been there. Today, we're going to break down a specific equation, 5x + (x + 3) = 38.4, and walk through the steps to solve it. Whether you're a student brushing up on your algebra skills or just someone who loves a good math challenge, this guide is for you. Let's dive in and make math a little less mysterious and a lot more fun!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what the equation 5x + (x + 3) = 38.4 is telling us. Equations like this are the backbone of algebra, and they represent a balance between two expressions. On one side of the equals sign, we have 5x + (x + 3), which is an algebraic expression involving a variable, x, and some constants. On the other side, we have 38.4, a constant value. Our goal is to find the value of x that makes the two sides equal.

Breaking it down further, the left side of the equation involves a combination of terms. We have 5x, which means 5 times the variable x, and (x + 3), which is an expression in parentheses indicating that we'll need to handle this part carefully. The number 3 is a constant term, meaning its value doesn't change. The plus signs indicate addition, so we're adding these terms together. Understanding each component helps us approach the problem systematically.

Why is understanding the equation so important? Well, think of it like reading a map before a journey. If you know your starting point and the terrain you'll be crossing, you're much more likely to reach your destination efficiently. In the same way, understanding the different parts of the equation—the variables, constants, and operations—gives you a clear roadmap for solving it. It's not just about memorizing steps; it's about grasping the underlying concepts. This deeper understanding makes math less intimidating and more intuitive. So, with our map in hand, let's start our journey towards finding the value of x!

Step-by-Step Solution

Okay, guys, let's get down to business and solve this equation! We're going to tackle 5x + (x + 3) = 38.4 step-by-step, so you can see exactly how it's done. Trust me, once you break it down, it's not as scary as it looks.

Step 1: Simplify the Equation

The first thing we want to do is simplify the equation. This means getting rid of any parentheses and combining like terms. Our equation is 5x + (x + 3) = 38.4. Notice the parentheses around (x + 3)? Since there's just a plus sign in front of them, we can simply remove them without changing anything. This gives us:

5x + x + 3 = 38.4

Now, we have some like terms on the left side: 5x and x. Like terms are terms that have the same variable raised to the same power. In this case, both terms have x raised to the power of 1 (which is usually not written explicitly). So, we can combine them. Think of it like having 5 apples and adding another apple—you end up with 6 apples. Similarly, 5x + x becomes 6x. Our equation now looks like this:

6x + 3 = 38.4

See? We've already made it simpler! Simplifying is crucial because it reduces the clutter and makes the equation easier to work with. It's like decluttering your workspace before starting a project—you'll feel much more organized and ready to go.

Step 2: Isolate the Variable Term

Next up, we need to isolate the variable term. In our equation, 6x + 3 = 38.4, the variable term is 6x. To isolate it, we want to get it all by itself on one side of the equation. This means we need to get rid of that + 3. Remember, equations are like a balancing act—what you do to one side, you have to do to the other to keep it balanced.

So, to get rid of the + 3, we'll do the opposite operation: subtraction. We'll subtract 3 from both sides of the equation:

6x + 3 - 3 = 38.4 - 3

This simplifies to:

6x = 35.4

Now, the variable term 6x is isolated on the left side. We're one step closer to finding the value of x. Isolating the variable term is a fundamental step in solving equations. It's like clearing a path to your destination, making it easier to see where you need to go next.

Step 3: Solve for the Variable

Alright, we're in the home stretch! We've simplified the equation and isolated the variable term. Now, all that's left is to solve for x. Our equation is currently 6x = 35.4. This means 6 times x equals 35.4. To find x, we need to undo the multiplication.

The opposite of multiplication is division, so we'll divide both sides of the equation by 6. This will get x all by itself:

(6x) / 6 = 35.4 / 6

On the left side, the 6s cancel each other out, leaving us with just x. On the right side, 35.4 divided by 6 is 5.9. So, we have:

x = 5.9

And there you have it! We've solved the equation. The value of x that makes the equation 5x + (x + 3) = 38.4 true is 5.9. Solving for the variable is the ultimate goal, and it's super satisfying when you finally get there. It's like reaching the summit after a long climb—you can look back and see all the steps you took to get there.

Verification

Before we pat ourselves on the back and move on, it's always a good idea to verify our solution. Verification is like double-checking your work—it ensures that you didn't make any mistakes along the way. To verify our solution, we'll plug the value we found for x (which is 5.9) back into the original equation and see if it holds true.

Our original equation is 5x + (x + 3) = 38.4. Let's substitute x with 5.9:

5(5.9) + (5.9 + 3) = 38.4

Now, we'll perform the calculations step by step. First, multiply 5 by 5.9:

29.5 + (5.9 + 3) = 38.4

Next, add 5.9 and 3 inside the parentheses:

29.5 + 8.9 = 38.4

Finally, add 29.5 and 8.9:

38.4 = 38.4

Look at that! The left side of the equation equals the right side. This means our solution, x = 5.9, is correct. Verification gives you confidence in your answer. It's like having a map to confirm you've reached the right destination. It's an essential step in problem-solving, not just in math but in many areas of life. So, always take the time to double-check your work—it's worth the effort!

Conclusion

Great job, guys! We've successfully solved the equation 5x + (x + 3) = 38.4. We took it step-by-step, from simplifying the equation to isolating the variable and finally solving for x. We even verified our answer to make sure we got it right. Remember, the key to tackling algebraic equations is to break them down into manageable steps and approach them systematically. Each step builds on the previous one, leading you closer to the solution.

Solving equations like this isn't just about getting the right answer; it's about building problem-solving skills that can be applied in many different situations. Whether you're balancing your budget, planning a project, or just trying to figure out how long it will take to get somewhere, the ability to think logically and solve problems is invaluable. So, keep practicing, keep exploring, and keep challenging yourself. Math can be a fascinating journey, and with each equation you solve, you're sharpening your mind and expanding your horizons. Until next time, happy problem-solving!