Solving 3x/4 = 2x + 5 A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks a bit intimidating? Don't worry, we've all been there! Today, we're going to break down a specific equation: 3x/4 = 2x + 5. This might seem complex at first, but trust me, with a step-by-step approach, it's totally manageable. We'll walk through each stage, explaining the logic and the math behind it. So, grab your pencils and let's dive in!

Understanding the Basics of Linear Equations

Before we jump into solving our equation, let's quickly recap what we're dealing with. The equation 3x/4 = 2x + 5 is a linear equation. In simple terms, a linear equation is an equation where the highest power of the variable (in this case, 'x') is 1. These equations, when graphed, produce a straight line – hence the name "linear." Our goal here is to find the value of 'x' that makes the equation true. Think of it like a puzzle: we need to figure out what 'x' is so that both sides of the equation are equal. The fundamental principle we'll use is that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain balance. This keeps the equation valid and helps us isolate 'x'. For example, if we add 5 to the left side, we must add 5 to the right side as well. This might seem straightforward, but it’s the golden rule of equation solving. Mastering this concept is crucial not just for solving this particular equation but for tackling any linear equation you might encounter. Keep in mind, the 'x' in the equation represents an unknown quantity, and our mission is to uncover it through a series of logical steps. So, with these basics in mind, let's get started on solving 3x/4 = 2x + 5.

Step 1: Eliminating the Fraction

Okay, so the first thing that might catch your eye in the equation 3x/4 = 2x + 5 is the fraction. Fractions can sometimes make equations look a bit messier, so let's get rid of it right away! The best way to eliminate a fraction in an equation is to multiply both sides of the equation by the denominator of that fraction. In our case, the denominator is 4. So, we're going to multiply both 3x/4 and (2x + 5) by 4. Remember that golden rule we talked about? What we do to one side, we must do to the other! This ensures the equation remains balanced. When we multiply 3x/4 by 4, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with just 3x. On the other side, we need to multiply both terms inside the parentheses (2x and 5) by 4. This is where the distributive property comes in handy. We multiply 4 by 2x, which gives us 8x, and then we multiply 4 by 5, which gives us 20. So, the right side of the equation becomes 8x + 20. After this step, our equation looks much cleaner: 3x = 8x + 20. We've successfully eliminated the fraction and taken a significant step toward isolating 'x'. This is a common and powerful technique in equation solving, and you'll find it incredibly useful in many situations. So, remember, when you see a fraction in an equation, your first instinct should be to multiply both sides by the denominator. Now that we've cleared the fraction, let's move on to the next step!

Step 2: Gathering the 'x' Terms

Now that we've got rid of the fraction, our equation looks like this: 3x = 8x + 20. The next step is to gather all the terms containing 'x' on one side of the equation. This is a crucial step in isolating 'x' and getting closer to our solution. Looking at our equation, we have 'x' terms on both sides – 3x on the left and 8x on the right. To consolidate them, we need to move one of these terms to the other side. The general strategy is to move the smaller 'x' term to the side with the larger 'x' term or to the side that has other constants. In our case, it’s easier to subtract 3x from both sides of the equation. This will eliminate the 'x' term on the left side and combine it with the 'x' term on the right side. When we subtract 3x from the left side (3x), we get 0. On the right side, we subtract 3x from 8x, which leaves us with 5x. The + 20 remains as it is because we haven't done anything to it yet. So, after this step, our equation transforms to 0 = 5x + 20. Notice how we've successfully grouped the 'x' terms on one side. This makes the equation simpler and easier to work with. Gathering like terms is a fundamental technique in algebra, and it's something you'll use frequently when solving equations. By doing this, we're essentially simplifying the equation and bringing it closer to a form where we can easily isolate 'x'. So, always remember to gather your 'x' terms together – it's a key step in the equation-solving process. Now, let's move on to the next step and continue our quest to find the value of 'x'!

Step 3: Isolating the Constant Term

Alright, we've made good progress! Our equation now stands at 0 = 5x + 20. Our next goal is to isolate the term with 'x' further. To do this, we need to get rid of the constant term on the same side as the 'x' term. In this case, the constant term is +20. To eliminate it, we'll do the opposite operation – we'll subtract 20 from both sides of the equation. Remember, balance is key! Whatever we do to one side, we must do to the other. When we subtract 20 from the left side (0), we get -20. On the right side, we subtract 20 from 5x + 20. The +20 and -20 cancel each other out, leaving us with just 5x. So, our equation now looks like this: -20 = 5x. See how we've managed to isolate the term with 'x' on one side of the equation? This is a significant step forward. By isolating the constant term, we're essentially peeling away the layers surrounding 'x', bringing us closer to revealing its value. This step is all about strategic manipulation – using inverse operations to move terms around and simplify the equation. Isolating terms is a fundamental skill in algebra, and it's essential for solving a wide range of equations. So, keep practicing this technique, and you'll become a pro at maneuvering terms and simplifying equations. Now that we've isolated the 'x' term, we're just one step away from finding our solution. Let's move on to the final step and crack this equation!

Step 4: Solving for 'x'

Okay, we're in the home stretch! Our equation is now -20 = 5x. We've done a fantastic job of simplifying it, and now it's time for the final step: solving for 'x'. To isolate 'x' completely, we need to get rid of the coefficient that's multiplying it. In this case, the coefficient is 5. The operation connecting 5 and 'x' is multiplication, so to undo it, we'll use the inverse operation: division. We're going to divide both sides of the equation by 5. Remember the golden rule – what we do to one side, we must do to the other! When we divide the left side (-20) by 5, we get -4. On the right side, we divide 5x by 5. The 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with just 'x'. So, after this step, our equation becomes -4 = x. And there you have it! We've solved for 'x'. The value of 'x' that makes the equation 3x/4 = 2x + 5 true is -4. Congratulations! You've successfully navigated the equation and found the solution. This final step is all about bringing it home – using the appropriate inverse operation to isolate the variable and reveal its value. Solving for 'x' is the ultimate goal in many algebraic problems, and you've just demonstrated your ability to do it. Remember, practice makes perfect, so keep tackling equations and honing your skills. Now that we've solved for 'x', let's take a moment to recap the entire process and reinforce what we've learned.

Summary of Steps

Let's do a quick rewind and recap the steps we took to solve the equation 3x/4 = 2x + 5. This will help solidify your understanding and make sure you're comfortable with the process. First, we eliminated the fraction by multiplying both sides of the equation by 4. This transformed our equation from 3x/4 = 2x + 5 to 3x = 8x + 20. Remember, getting rid of fractions often makes equations much easier to handle. Next, we gathered the 'x' terms on one side of the equation. We subtracted 3x from both sides, which gave us 0 = 5x + 20. Grouping like terms is a key strategy in simplifying equations. Then, we isolated the constant term by subtracting 20 from both sides. This resulted in -20 = 5x. Isolating terms helps us peel away the layers and get closer to the variable we're trying to solve for. Finally, we solved for 'x' by dividing both sides by 5. This gave us our solution: x = -4. This final step is where we reveal the value of 'x' and complete the puzzle. By recapping these steps, you can see the logical progression we followed to solve the equation. Each step built upon the previous one, leading us to our final answer. Remember, equation solving is like a journey – each step is a milestone that brings you closer to your destination. So, keep practicing these steps, and you'll become a confident equation solver! Now, let's move on to discussing how we can verify our solution to make sure it's correct.

Verifying the Solution

So, we've found that x = -4 is the solution to the equation 3x/4 = 2x + 5. But how can we be absolutely sure that our answer is correct? That's where verification comes in! Verifying our solution is like double-checking our work – it gives us peace of mind that we've done everything right. The process is simple: we substitute the value we found for 'x' (in this case, -4) back into the original equation and see if both sides of the equation are equal. If they are, then our solution is correct! Let's walk through it. Our original equation is 3x/4 = 2x + 5. We're going to replace every 'x' in the equation with -4. So, the left side becomes 3(-4)/4*, and the right side becomes 2(-4) + 5*. Now, let's simplify each side separately. On the left side, 3(-4)* is -12, and -12/4 is -3. So, the left side simplifies to -3. On the right side, 2(-4)* is -8, and -8 + 5 is -3. So, the right side also simplifies to -3. We've found that both sides of the equation equal -3 when we substitute x = -4. This means our solution is correct! Verification is a crucial step in equation solving, and it's something you should always do. It not only confirms your answer but also helps you catch any mistakes you might have made along the way. So, remember, once you've solved an equation, take a few extra moments to verify your solution – it's worth the effort! Now that we've verified our solution, let's wrap things up with some final thoughts and takeaways.

Conclusion and Key Takeaways

Awesome job, guys! We've successfully solved the equation 3x/4 = 2x + 5 and verified our solution. We started with what might have seemed like a complex equation, but by breaking it down into manageable steps, we were able to find the value of 'x'. Let's recap some of the key takeaways from this exercise. First, we learned the importance of eliminating fractions by multiplying both sides of the equation by the denominator. This simplifies the equation and makes it easier to work with. Second, we practiced gathering like terms, specifically the 'x' terms, on one side of the equation. This is a fundamental technique in algebra and helps us isolate the variable. Third, we mastered the skill of isolating the variable by using inverse operations. We subtracted and divided to get 'x' by itself. Finally, we emphasized the crucial step of verifying our solution by substituting it back into the original equation. This ensures that our answer is correct and gives us confidence in our work. Solving equations is a fundamental skill in mathematics, and it's something you'll use in many different contexts. The key is to break down complex problems into smaller, more manageable steps. By following a systematic approach and remembering the golden rule of balance, you can tackle any equation that comes your way. So, keep practicing, keep exploring, and keep building your equation-solving skills. You've got this! Remember, math is like a muscle – the more you use it, the stronger it gets. So, keep flexing those mathematical muscles, and you'll become a true equation-solving champion!