Solving The Equation 3(5x+2)/2 = X+1: A Step-by-Step Guide

Hey guys! Let's dive into solving this equation together. Mathematics can sometimes seem like a maze, but with a systematic approach, we can conquer any problem. In this comprehensive guide, we'll break down the equation $\frac{3(5x+2)}{2} = x+1$ step-by-step, ensuring you not only understand the solution but also the underlying principles. So, buckle up and get ready to sharpen your math skills! Understanding the fundamentals of algebra is crucial, so let's make sure we cover those bases. Remember, practice makes perfect, and the more you engage with these concepts, the more confident you'll become. We'll use various strategies, including distribution, simplification, and isolating the variable, to arrive at the solution. This process is applicable to a wide range of algebraic equations, making this a valuable skill to master. Think of each step as a piece of the puzzle, and by the end, you'll have a clear picture of how to solve equations like this one. Let's begin by understanding the initial equation and the operations involved. We'll then proceed to simplify it, step-by-step, until we find the value of x. Don't worry if some parts seem tricky at first; we'll explain everything in detail and provide examples to help you along the way. Ready? Let’s jump in!

Step 1: Distribute the 3 in the numerator

To kick things off, we need to simplify the left side of the equation. The first thing we'll do is distribute the 3 in the numerator: $\frac{3(5x+2)}{2}$. This means we're going to multiply 3 by both terms inside the parentheses: $5x$ and $2$. Distribution is a fundamental algebraic technique that allows us to handle expressions enclosed in parentheses. Think of it like sharing – the number outside the parentheses gets multiplied with each term inside. It's essential to get this step right because any mistake here will affect the rest of the solution. So, let's break it down. 3 multiplied by $5x$ is $15x$, and 3 multiplied by 2 is 6. This gives us a new numerator: $15x + 6$. So, our equation now looks like this: $\frac{15x + 6}{2} = x + 1$. See how we've already made progress? By distributing the 3, we've eliminated the parentheses and made the equation a bit easier to handle. This step sets the stage for the next steps, where we'll continue to simplify and isolate the variable x. Remember, each step builds on the previous one, so understanding this initial distribution is crucial. Now, let’s move on to the next step, where we'll eliminate the fraction and further simplify the equation. Keep up the great work, guys! We're one step closer to solving it!

Step 2: Multiply both sides by 2 to eliminate the fraction

Now that we've distributed the 3, let’s tackle that fraction! Fractions can sometimes make equations look more intimidating than they are, but don't worry, we've got a way to get rid of it. The denominator on the left side is 2, so to eliminate it, we're going to multiply both sides of the equation by 2. This is a crucial step because it simplifies the equation significantly, making it much easier to work with. Remember, what we do to one side of the equation, we must do to the other to maintain balance – a fundamental principle in algebra. So, let's multiply both $\frac{15x + 6}{2}$ and $x + 1$ by 2. On the left side, multiplying by 2 cancels out the denominator, leaving us with just the numerator: $15x + 6$. On the right side, we need to distribute the 2 to both terms: 2 multiplied by x is $2x$, and 2 multiplied by 1 is 2. So, the right side becomes $2x + 2$. Our equation now looks like this: $15x + 6 = 2x + 2$. Look at how much simpler it is! By eliminating the fraction, we've made the equation much more manageable. This step highlights the importance of using inverse operations to simplify equations. Multiplication is the inverse of division, so multiplying by 2 effectively undoes the division by 2 on the left side. Now that we have a clean, linear equation, we're ready to move on to the next step: isolating the variable x. Keep going, you're doing awesome!

Step 3: Move the x terms to one side

Alright, we're making great progress! We've eliminated the fraction and now have a much cleaner equation: $15x + 6 = 2x + 2$. The next step is to gather all the x terms on one side of the equation. This is a common strategy in solving algebraic equations, as it helps us isolate the variable we're trying to find. To do this, we're going to subtract $2x$ from both sides of the equation. Why subtract $2x$? Because it's the opposite operation of adding $2x$, and we want to eliminate it from the right side. Remember, maintaining balance is key, so whatever we do to one side, we must do to the other. When we subtract $2x$ from the left side ($15x + 6$), we get $15x - 2x + 6$, which simplifies to $13x + 6$. On the right side, subtracting $2x$ from $2x + 2$ leaves us with just 2, since $2x - 2x = 0$. So, our equation now looks like this: $13x + 6 = 2$. See how we're slowly but surely isolating the x term? By moving the x terms to one side, we've simplified the equation further and made it easier to solve. This step demonstrates the power of using inverse operations to manipulate equations. Subtraction is the inverse of addition, so subtracting $2x$ helps us eliminate it from one side of the equation. Now that we have all the x terms on one side, we can move on to the next step: isolating the x term completely. You're doing fantastic! Let’s keep up the momentum!

Step 4: Isolate the x term

We're on the home stretch now! Our equation currently looks like this: $13x + 6 = 2$. We've successfully gathered the x terms on one side, and now it's time to isolate the x term completely. This means we need to get rid of the +6 on the left side. To do this, we'll use the inverse operation of addition, which is subtraction. We're going to subtract 6 from both sides of the equation. Again, remember the golden rule: whatever you do to one side, you must do to the other to maintain balance. When we subtract 6 from the left side ($13x + 6$), the +6 and -6 cancel each other out, leaving us with just $13x$. On the right side, subtracting 6 from 2 gives us $2 - 6$, which equals -4. So, our equation now looks like this: $13x = -4$. Look at how much simpler it is! We've successfully isolated the x term on one side of the equation. This step highlights the importance of using inverse operations strategically. Subtraction is the inverse of addition, and by subtracting 6, we've effectively removed it from the left side. Now that we have $13x = -4$, we're just one step away from finding the value of x. All that's left to do is divide both sides by 13. Are you ready to finish this? Let’s go for it!

Step 5: Solve for x

We've reached the final step! Our equation is now $13x = -4$. To solve for x, we need to isolate it completely. Currently, x is being multiplied by 13, so to undo this multiplication, we'll use the inverse operation: division. We're going to divide both sides of the equation by 13. Remember, maintaining balance is crucial, so whatever we do to one side, we must do to the other. When we divide the left side ($13x$) by 13, the 13s cancel each other out, leaving us with just x. On the right side, we divide -4 by 13, which gives us $-\frac{4}{13}$. So, our solution is $x = -\frac{4}{13}$. Congratulations, we've solved the equation! By dividing both sides by 13, we've successfully isolated x and found its value. This final step underscores the importance of using inverse operations to unravel equations. Division is the inverse of multiplication, and by dividing by 13, we've effectively undone the multiplication and revealed the value of x. You've now seen the entire process, from distributing to eliminating fractions to isolating the variable. Remember, practice is key to mastering these skills. The more you work through equations like this, the more confident you'll become. Great job, guys! You nailed it!

Final Answer

So, the final answer to the equation $\frac{3(5x+2)}{2} = x+1$ is $x = -\frac{4}{13}$. We've walked through each step carefully, from distributing and eliminating fractions to isolating the variable and solving for x. Remember, math might seem daunting at first, but with a systematic approach and a little practice, you can tackle any equation that comes your way. Keep up the great work, and always remember to double-check your answers! If you ever feel stuck, revisit these steps and break the problem down into smaller, more manageable parts. And hey, don't hesitate to ask for help when you need it. Math is a journey, and we're all in it together. You've got this!