Solving The Equation: 4x - 3 + 5x = 15 - A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks a bit intimidating at first glance? Don't worry, we've all been there! Today, we're going to break down a common type of algebraic equation and solve it together. Our mission? To conquer the equation 4x - 3 + 5x = 15. This might seem tricky, but I promise it’s totally manageable once you understand the steps. So, grab your pencils, and let's dive in!

Understanding the Basics of Algebraic Equations

Before we jump into the specifics of this equation, let's quickly recap some fundamental concepts. At its heart, an algebraic equation is a mathematical statement that shows the equality between two expressions. It typically involves variables (like our 'x'), constants (numbers), and operations (addition, subtraction, multiplication, division). The goal is always to find the value of the variable that makes the equation true.

In our case, the variable is 'x,' and we need to figure out what number 'x' represents so that when we plug it into the equation, both sides are equal. Think of it like a puzzle where we need to find the missing piece. The beauty of algebra is that there are clear, logical steps to follow to solve these puzzles. Equations are the backbone of mathematics and are crucial in various fields, including physics, engineering, and even economics. Knowing how to solve them opens doors to understanding more complex concepts and problem-solving scenarios.

When tackling equations, remember the golden rule: whatever you do to one side, you must do to the other. This ensures that the equality remains balanced. For example, if we add a number to the left side, we must add the same number to the right side. This principle will be key as we work through our equation. So, with the basics in mind, let’s roll up our sleeves and get to work on solving 4x - 3 + 5x = 15. We'll break it down step by step, making it super easy to follow!

Step 1: Combine Like Terms

Alright, let's get started! The first thing we want to do when solving the equation 4x - 3 + 5x = 15 is to simplify each side as much as possible. This often involves combining 'like terms.' Now, what exactly are like terms? Simply put, like terms are terms that have the same variable raised to the same power. In our equation, we have two terms with 'x': 4x and 5x. These are our like terms, and we can combine them.

Think of 'x' as a tangible object, like an apple. So, 4x means we have 4 apples, and 5x means we have 5 apples. If we add them together, we have a total of 9 apples. Mathematically, this looks like: 4x + 5x = 9x. Super straightforward, right? Now our equation looks a little cleaner: 9x - 3 = 15. We’ve successfully combined the 'x' terms, making the equation less cluttered and easier to work with. This step is crucial because it reduces the complexity of the equation and brings us closer to isolating the variable.

But why do we need to combine like terms in the first place? Well, it's all about efficiency. By combining terms, we reduce the number of operations we need to perform, which simplifies the overall process. Imagine trying to solve a really long equation without combining like terms – it would be a total mess! So, this first step is like tidying up before you start a big project. It makes everything more organized and manageable. Trust me, getting good at this step will make solving equations a whole lot smoother. Now that we've combined like terms, let's move on to the next step and continue our journey to find the value of 'x'.

Step 2: Isolate the Variable Term

Now that we've combined like terms and our equation looks like 9x - 3 = 15, it's time to isolate the variable term. What does this mean? Essentially, we want to get the term with 'x' (in this case, 9x) all by itself on one side of the equation. To do this, we need to get rid of the '-3' that's hanging out on the left side. Remember our golden rule: whatever we do to one side, we must do to the other.

The opposite of subtracting 3 is adding 3. So, let's add 3 to both sides of the equation. This looks like: 9x - 3 + 3 = 15 + 3. On the left side, the -3 and +3 cancel each other out, leaving us with just 9x. On the right side, 15 + 3 equals 18. So, our equation now reads: 9x = 18. Woohoo! We've made some significant progress. We’ve successfully isolated the term with 'x' on one side, making it much clearer what our next step should be. Isolating the variable term is a crucial step in solving equations because it brings us closer to figuring out the value of 'x'.

By strategically using inverse operations (like adding 3 to undo subtracting 3), we can systematically peel away the extra numbers surrounding our variable. Think of it like unwrapping a present – each step gets us closer to the treasure inside. This process might seem simple, but it's a fundamental technique in algebra. Mastering it will empower you to tackle more complex equations with confidence. So, with our variable term nicely isolated, we’re ready to take the final plunge and solve for 'x'. Let’s head on to the next step and seal the deal!

Step 3: Solve for the Variable

We're in the home stretch! Our equation is now in the form 9x = 18. This is fantastic because we’re just one step away from solving for 'x.' Remember, 9x means 9 multiplied by x. So, to isolate 'x' completely, we need to undo this multiplication. And what’s the opposite of multiplication? You guessed it – division!

To get 'x' by itself, we'll divide both sides of the equation by 9. This looks like: (9x) / 9 = 18 / 9. On the left side, the 9s cancel each other out, leaving us with just 'x.' On the right side, 18 divided by 9 is 2. So, our final answer is: x = 2. Boom! We did it!

We've successfully solved the equation. This final step is all about applying the inverse operation to completely isolate the variable. By dividing both sides by the coefficient (the number multiplied by the variable), we reveal the true value of 'x.' This step is like the grand finale of our equation-solving journey. Once we've isolated the variable, we've essentially cracked the code and found the solution. But don't just take our word for it – it's always a good idea to check your work!

Checking Your Solution

Now, before we declare victory, let’s make absolutely sure our answer is correct. The best way to do this is to substitute our solution (x = 2) back into the original equation and see if it holds true. Our original equation was 4x - 3 + 5x = 15. Let's plug in x = 2:

4*(2) - 3 + 5*(2) = 15 8 - 3 + 10 = 15 5 + 10 = 15 15 = 15

It checks out! Both sides of the equation are equal when x = 2. This means our solution is correct. Yay for us! Checking your solution is a vital step in problem-solving. It gives you confidence that you've not only found an answer but that it's the right one. Think of it as the final seal of approval on your mathematical masterpiece.

Conclusion: You've Got This!

So, there you have it! We've successfully solved the equation 4x - 3 + 5x = 15, and we found that x = 2. We walked through each step, from combining like terms to isolating the variable and finally solving for 'x.' Remember, the key to solving algebraic equations is to break them down into smaller, manageable steps. Start by simplifying, then isolate, and finally solve.

And don't forget to check your work! This will help you catch any errors and build your confidence. Guys, algebra might seem intimidating at first, but with practice and a clear understanding of the basic steps, you can conquer any equation that comes your way. Keep practicing, keep asking questions, and keep exploring the awesome world of mathematics. You've got this!