Solving (2/3)x + 3 = (1/4)x - 7: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of algebra to tackle a classic equation: (2/3)x + 3 = (1/4)x - 7. If you've ever felt a little intimidated by fractions and variables, don't worry! We're going to break this down step-by-step, making it super easy to understand. Think of this as your friendly guide to conquering algebraic equations. Let’s get started and unravel this mathematical puzzle together!
Understanding the Equation
Before we jump into solving, let's make sure we understand what this equation is all about. At its core, this equation is a statement of equality. We're saying that the expression on the left side, (2/3)x + 3, is equal to the expression on the right side, (1/4)x - 7. Our mission is to find the value of 'x' that makes this statement true. In simpler terms, we want to figure out what number 'x' needs to be so that both sides of the equation balance perfectly. This is the fundamental goal of solving any algebraic equation, and once you grasp this concept, you're already halfway there! It's like a balancing scale – we need to keep both sides equal as we manipulate the equation to isolate 'x'.
Identifying the Components
Let’s break down the components of the equation to make it even clearer. We have 'x', which is our variable – the unknown value we're trying to find. Then we have coefficients, which are the numbers multiplied by 'x': 2/3 and 1/4. We also have constants, which are the standalone numbers: 3 and -7. Understanding these components is crucial because each plays a role in how we solve the equation. For instance, the fractions might seem a bit scary at first, but we'll learn how to deal with them effectively. Recognizing these parts helps us strategize our approach. We'll see how to combine like terms, move constants to one side, and eventually isolate 'x' to find its value. Remember, algebra is like a puzzle, and identifying the pieces is the first step to solving it.
Why is this Important?
You might be wondering, “Why bother solving equations like this?” Well, algebra is a foundational concept in mathematics and has countless real-world applications. Solving equations helps us model and understand various situations, from calculating finances to designing structures. Think about it: engineers use equations to determine the strength of bridges, economists use them to predict market trends, and even video game developers use them to create realistic physics. So, mastering these skills isn't just about getting good grades; it's about building a toolkit for problem-solving in all sorts of fields. Plus, the logical thinking and analytical skills you develop by solving equations will benefit you in many aspects of life, from making informed decisions to tackling everyday challenges. So, let's dive in and equip ourselves with these powerful tools!
Step 1: Eliminate the Fractions
Okay, first things first, let's tackle those fractions! Fractions can sometimes make equations look more complicated than they actually are, but we have a neat trick to get rid of them. The key is to find the least common multiple (LCM) of the denominators. In our equation, (2/3)x + 3 = (1/4)x - 7, the denominators are 3 and 4. So, what's the LCM of 3 and 4? If you're thinking 12, you're spot on! The LCM is the smallest number that both 3 and 4 can divide into evenly.
Multiplying by the LCM
Now, here's the magic step: we're going to multiply every term in the equation by the LCM, which is 12. This means we'll multiply (2/3)x by 12, 3 by 12, (1/4)x by 12, and -7 by 12. Why do we do this? Because multiplying by the LCM will cancel out the denominators, leaving us with whole numbers. It's like a mathematical clean-up operation! Let’s see how this looks:
- 12 * (2/3)x = (12 * 2) / 3 * x = 24 / 3 * x = 8x
- 12 * 3 = 36
- 12 * (1/4)x = (12 * 1) / 4 * x = 12 / 4 * x = 3x
- 12 * (-7) = -84
See how the fractions disappeared? Our equation now looks much simpler: 8x + 36 = 3x - 84. This is a huge step forward because we've transformed a potentially messy equation into something much more manageable. Getting rid of fractions often makes the rest of the solving process smoother and less prone to errors. So, remember this trick – it's a lifesaver when dealing with equations that have fractions!
The New Equation
After multiplying each term by the LCM, we've successfully eliminated the fractions. Our equation has transformed from (2/3)x + 3 = (1/4)x - 7 into the much cleaner and easier-to-handle form of 8x + 36 = 3x - 84. This new equation is equivalent to the original one, meaning it has the same solution for 'x'. The only difference is that it's free of fractions, which simplifies the subsequent steps. This is a common strategy in algebra: we manipulate equations to make them easier to solve without changing their fundamental truth. Now that we have a friendlier equation, we can move on to the next step: grouping like terms. This involves bringing all the 'x' terms to one side and all the constant terms to the other, setting us up to finally isolate 'x' and find its value. So, we've made great progress – let's keep going!
Step 2: Group Like Terms
Alright, now that we've waved goodbye to those pesky fractions, let's focus on grouping like terms. What does that mean, exactly? Well, in our equation 8x + 36 = 3x - 84, we have terms with 'x' (8x and 3x) and constant terms (36 and -84). Our goal here is to get all the 'x' terms on one side of the equation and all the constants on the other side. This makes the equation much easier to solve because we're essentially organizing the information. Think of it like sorting your clothes – you put all the shirts in one pile and all the pants in another. It’s the same idea with algebraic terms!
Moving the 'x' Terms
Let's start by moving the 'x' terms. We have 8x on the left side and 3x on the right side. A common strategy is to move the smaller 'x' term to the side with the larger 'x' term. In this case, we'll move 3x from the right side to the left side. How do we do that? We subtract 3x from both sides of the equation. Remember, whatever we do to one side of the equation, we have to do to the other to keep the equation balanced. So:
8x + 36 - 3x = 3x - 84 - 3x
This simplifies to:
5x + 36 = -84
See what happened? We subtracted 3x from both sides, and now we have 5x on the left side. We're one step closer to isolating 'x'! This process of adding or subtracting terms from both sides is a fundamental technique in algebra. It allows us to rearrange the equation without changing its solution. We're essentially shifting terms around to get them where we want them, making the equation more manageable. Now, let's move on to the constants and get them on the other side.
Moving the Constants
Now that we've grouped the 'x' terms on the left side, let's move the constants to the right side. We have the constant 36 on the left and -84 on the right. To move 36 to the right side, we'll subtract 36 from both sides of the equation. This is the same principle we used for the 'x' terms – we're doing the same operation on both sides to maintain balance. So:
5x + 36 - 36 = -84 - 36
This simplifies to:
5x = -120
Fantastic! We've successfully grouped the constants on the right side. Our equation now reads 5x = -120. Look how much simpler it has become! We've gone from a complex equation with fractions and multiple terms to a very straightforward one. This is the power of grouping like terms – it distills the equation down to its essence, making it much easier to see the path to the solution. Now, we're just one step away from finding the value of 'x'. We have 5x equals -120, and all that's left to do is isolate 'x' completely.
Step 3: Isolate the Variable
We're in the home stretch now! We've eliminated the fractions, grouped like terms, and our equation is looking super clean: 5x = -120. The final step is to isolate 'x', which means getting 'x' all by itself on one side of the equation. To do this, we need to undo the operation that's being performed on 'x'. In this case, 'x' is being multiplied by 5. So, to undo that multiplication, we'll do the opposite operation: division. We're going to divide both sides of the equation by 5.
Dividing Both Sides
Here's how it looks:
(5x) / 5 = -120 / 5
On the left side, the 5s cancel each other out, leaving us with just 'x'. On the right side, -120 divided by 5 is -24. So, our equation simplifies to:
x = -24
And there you have it! We've successfully isolated 'x' and found its value. This is the moment of truth in solving any algebraic equation – the point where we discover the number that makes the equation balance. Dividing both sides by the coefficient of 'x' is the standard way to isolate the variable, and it's a technique you'll use again and again in algebra. Now that we have our solution, let's take a moment to celebrate our success and then verify that our answer is correct.
The Solution
After all our hard work, we've arrived at the solution: x = -24. This means that if we substitute -24 for 'x' in the original equation, both sides should be equal. It's like finding the missing piece of a puzzle – we've discovered the value that makes the equation whole. Solving for 'x' is the core goal of this type of problem, and we've accomplished it by systematically simplifying the equation and isolating the variable. But before we declare victory, it's always a good idea to check our work. This ensures that we haven't made any mistakes along the way and that our solution is indeed correct. So, let's move on to the final step: verifying our solution.
Step 4: Verify the Solution
Okay, we've got our answer: x = -24. But before we pat ourselves on the back, let's make absolutely sure it's correct. How do we do that? We plug our solution back into the original equation. This is a crucial step because it confirms that our solution works and catches any potential errors we might have made along the way. Think of it as a final exam for our solution – it has to pass the test of the original equation!
Plugging it Back In
Our original equation was (2/3)x + 3 = (1/4)x - 7. Now, we're going to replace 'x' with -24:
(2/3) * (-24) + 3 = (1/4) * (-24) - 7
Let's simplify each side separately. On the left side:
(2/3) * (-24) = -16
-16 + 3 = -13
So, the left side simplifies to -13. Now, let's tackle the right side:
(1/4) * (-24) = -6
-6 - 7 = -13
The right side also simplifies to -13! Voila! Both sides of the equation are equal when x = -24. This means our solution is correct. We've successfully solved the equation and verified our answer. Plugging the solution back into the original equation is a powerful way to check your work and build confidence in your problem-solving skills. It's like having a built-in error detector that ensures you're on the right track.
The Final Check
We've reached the finish line, guys! We plugged our solution, x = -24, back into the original equation, (2/3)x + 3 = (1/4)x - 7, and we saw that both sides equal -13. This confirms that our solution is indeed correct. We've navigated through fractions, grouped like terms, isolated the variable, and verified our answer. This is a testament to the power of step-by-step problem-solving in algebra. Remember, each step we took was designed to simplify the equation and bring us closer to the solution. And now, we can confidently say that we've conquered this equation! Solving algebraic equations is a skill that builds over time with practice, so keep at it, and you'll become even more proficient. We did it!
Conclusion
So, there you have it! We've successfully solved the equation (2/3)x + 3 = (1/4)x - 7, and we found that x = -24. Along the way, we learned how to eliminate fractions, group like terms, isolate the variable, and verify our solution. These are fundamental skills in algebra, and mastering them will open doors to more complex mathematical concepts. Remember, the key to solving equations is to break them down into manageable steps and approach each step with confidence. Don't be afraid of fractions or negative numbers – with practice, they'll become your friends. Algebra is a powerful tool, and every equation you solve is a step towards building your problem-solving abilities. So, keep practicing, keep exploring, and keep enjoying the world of mathematics! You've got this!