Solving $3(x-4)-(3x-4)=6x$: A Step-by-Step Guide
Hey everyone! Today, we're diving deep into the mathematical realm to dissect and conquer the equation . This equation, at first glance, might seem a bit daunting, but fear not! We're going to break it down step-by-step, ensuring that each twist and turn is crystal clear. Our journey will not only solve for the elusive 'x' but also illuminate the fundamental principles of algebraic manipulation. So, grab your metaphorical math hats, and let's embark on this exciting adventure together!
Unraveling the Mystery: A Step-by-Step Guide
Step 1: The Art of Distribution
Our initial equation stands as . The first hurdle we encounter is the parentheses. To overcome this, we employ the distributive property. This property, a cornerstone of algebra, allows us to multiply the term outside the parentheses by each term inside. For the first set of parentheses, we distribute the '3' across '(x-4)'. This yields , which simplifies to . Now, let's tackle the second set of parentheses, '-(3x-4)'. Here, it's crucial to recognize that the negative sign acts as a '-1' multiplier. Distributing this '-1' across '(3x-4)' gives us , simplifying to . Remember, a negative times a negative results in a positive! Now, let's rewrite our equation with these expansions: . This transformation is pivotal, as it clears the initial complexity and sets the stage for combining like terms.
Step 2: Combining Like Terms - The Great Gathering
With the parentheses vanished, our equation now reads . The next strategic move is to gather and combine 'like terms'. Like terms are those that share the same variable raised to the same power. In our equation, we have two types of like terms: terms with 'x' ( and ) and constant terms (-12 and +4). Let's start with the 'x' terms. We have , which beautifully cancels each other out, resulting in '0'. This simplification is a delightful turn of events! Now, let's focus on the constant terms: -12 + 4. Combining these gives us -8. Our equation, after this meticulous combing, is dramatically simplified to -8 = 6x. This step showcases the elegance of algebraic manipulation, where strategic combination can lead to significant simplification.
Step 3: Isolating 'x' - The Quest for Solitude
Our equation has been refined to -8 = 6x. Now, the penultimate step in our mathematical journey is to isolate 'x'. Isolating a variable means getting it all by itself on one side of the equation. In our case, 'x' is currently being multiplied by '6'. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by '6'. This maintains the balance of the equation, a fundamental principle in algebra. Dividing both sides by '6' gives us -8 / 6 = 6x / 6. The '6' on the right side cancels out, leaving us with 'x' isolated. On the left side, -8 / 6 simplifies to -4 / 3. Thus, we arrive at . This moment of isolation is a triumphant one, as it brings us closer to our final solution.
Step 4: The Grand Finale - Expressing the Solution
After our diligent efforts, we've arrived at . This is the solution to our equation! However, the art of mathematics often involves expressing solutions in various forms. is a perfectly valid solution, but it's also beneficial to understand its representation as a mixed number. To convert to a mixed number, we divide -4 by 3. The quotient is -1, and the remainder is -1. Thus, can also be expressed as -1 rac{1}{3}. Both and -1 rac{1}{3} are accurate representations of our solution. The choice of which form to use often depends on the context or the specific requirements of the problem. We have now successfully decoded the equation, revealing the value of 'x'.
Delving Deeper: Grasping the Core Concepts
The Distributive Property: More Than Just Multiplication
The distributive property is a fundamental concept in algebra, and as we saw in the first step, it's crucial for simplifying expressions and solving equations. But what exactly is it, and why is it so important? At its core, the distributive property allows us to multiply a single term by two or more terms inside a set of parentheses. Mathematically, it's expressed as a(b + c) = ab + ac. This might seem like a simple formula, but its implications are vast. It's not just about multiplying numbers; it's about distributing the 'weight' of one term across a sum or difference.
Think of it like this: imagine you're buying 3 bags of fruit, each containing 2 apples and 4 oranges. Instead of counting each fruit individually, you can use the distributive property. You have 3 * (2 apples + 4 oranges), which is the same as (3 * 2 apples) + (3 * 4 oranges), resulting in 6 apples and 12 oranges. This simple analogy highlights the power of distribution in simplifying complex situations. In algebra, this property is indispensable for expanding expressions, combining like terms, and ultimately, solving equations. It's a tool that allows us to break down complex problems into manageable parts, making it a cornerstone of algebraic manipulation. Understanding and mastering the distributive property is like unlocking a secret passage in the world of mathematics, opening doors to more advanced concepts and problem-solving techniques. So, the next time you encounter parentheses in an equation, remember the power of distribution โ it's your key to simplification and success!
Combining Like Terms: The Art of Simplification
Combining like terms is another crucial skill in algebra, a process that simplifies equations and makes them easier to solve. But what exactly are 'like terms', and why is combining them so important? Like terms are terms that have the same variable raised to the same power. For example, and are like terms because they both contain the variable 'x' raised to the power of 1. Similarly, and are like terms because they both contain the variable 'y' raised to the power of 2. However, and are not like terms because, although they share the same variable 'x', the powers are different (1 and 2, respectively).
The importance of combining like terms lies in its ability to simplify complex expressions. When we combine like terms, we're essentially adding or subtracting their coefficients (the numbers in front of the variables). This process reduces the number of terms in an equation, making it more manageable and easier to solve. Think of it as decluttering your mathematical space โ by grouping similar items together, you create order and clarity. For instance, in the expression , we can combine the 'x' terms ( and ) to get , and the 'y' terms ( and ) to get . This simplifies the expression to , which is much cleaner and easier to work with. Mastering the art of combining like terms is like having a superpower in algebra โ it allows you to transform complex equations into simpler, more solvable forms. So, embrace this skill, and watch as your algebraic abilities soar!
Isolating the Variable: The Quest for 'x'
Isolating the variable is a fundamental technique in algebra, a crucial step in solving equations and uncovering the value of the unknown. But what does it mean to isolate a variable, and why is it so important? Isolating a variable means getting that variable all by itself on one side of the equation, with all other terms on the opposite side. This is the key to unlocking the variable's value, as it reveals what the variable is equal to. Think of it as a detective trying to isolate a suspect from a crowd โ the goal is to separate the individual from the group to get a clear view. In algebra, the variable is our suspect, and we use mathematical operations to isolate it.
The process of isolating a variable involves using inverse operations. Inverse operations are operations that 'undo' each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. If a variable is being added to a term, we subtract that term from both sides of the equation to isolate the variable. If a variable is being multiplied by a number, we divide both sides of the equation by that number to isolate the variable. The golden rule of equation solving is that whatever operation you perform on one side of the equation, you must perform on the other side to maintain balance. This ensures that the equation remains true. For instance, in the equation , we first subtract 3 from both sides to get , and then divide both sides by 2 to get . Isolating the variable is like finding the hidden treasure in a mathematical puzzle โ it's the moment when the solution is revealed. So, hone your skills in this technique, and you'll be well-equipped to conquer any equation that comes your way!
Real-World Applications: Math in Everyday Life
While algebra might seem abstract, its principles are woven into the fabric of our daily lives. The equation we've dissected, , might not directly pop up in a grocery store, but the underlying concepts certainly do. Let's explore some real-world applications where these algebraic skills shine.
Imagine you're planning a budget for a trip. You have a fixed amount of money and need to allocate it for various expenses like travel, accommodation, and activities. This is where algebraic thinking comes into play. You might have equations representing the total cost, where 'x' could be the amount you spend on each activity. By manipulating these equations, you can figure out how to optimize your spending and stay within budget. Similarly, in cooking, you often need to adjust recipes based on the number of servings you want to make. If a recipe calls for certain amounts of ingredients for 4 people, you can use algebraic proportions to scale it up or down for a different number of people. The distributive property, a key element in our equation, is also crucial in calculating discounts or sales tax. When you see a '20% off' sale, you're essentially using the distributive property to calculate the reduced price. Algebra is also fundamental in fields like engineering, physics, and computer science. Engineers use equations to design structures and systems, physicists use them to model the behavior of the universe, and computer scientists use them to develop algorithms and software. Even in seemingly simple tasks like calculating the distance you'll travel at a certain speed, you're implicitly using algebraic concepts. The formula distance = speed * time is a classic example of an algebraic equation. So, the next time you're faced with a problem, remember that the tools of algebra are at your disposal. They're not just abstract symbols and equations; they're powerful instruments for understanding and navigating the world around us. Math is not just confined to the classroom; it's a vital part of our everyday lives!
In conclusion, guys, we've not only solved the equation but also uncovered the beauty and practicality of algebra. Keep practicing, and you'll become mathematical maestros in no time!
