Solve & Verify: 2(3-4x)=3(5x+3) Linear Equation
Hey There, Math Enthusiasts! Understanding Linear Equations is Key!
What's up, everyone? Today, we're diving headfirst into the fantastic world of linear equations. And trust me, guys, it's not as scary as it sounds! In fact, mastering linear equations is like unlocking a secret superpower that helps you solve problems not just in math class, but in real life too. Think about it: figuring out how much change you'll get back, calculating travel time, or even balancing your budget – these all involve the fundamental principles of linear equations. Our mission today is to solve and check the equation 2(3-4x)=3(5x+3). This specific equation, while looking a bit gnarly at first glance with all those numbers and parentheses, is a perfect example of a typical linear equation you'll encounter. We're going to break it down, step by logical step, so you can see exactly how to approach these kinds of problems with confidence. No more guessing, no more panicking! We'll cover everything from getting rid of those pesky parentheses using the distributive property to gathering all your 'x' terms and finally, isolating 'x' to find its true value. But we won't stop there! A true math pro always verifies their work. So, after we've solved it, we'll dive right back in and check our answer to make sure both sides of the equation perfectly balance out. This entire process is crucial for developing strong algebraic skills, which are foundational for more advanced mathematics and countless practical applications. So, grab your imaginary (or real!) calculator, a pencil, and some scratch paper, because we're about to make this equation our absolute bitch! Get ready to boost your algebraic game and feel like a total whiz kid. This guide is all about empowering you to tackle these challenges confidently and effectively. We're going to ensure you not only understand how to solve it, but why each step is taken, making you a master of linear equations.
The Grand Challenge: Unpacking Our Equation 2(3-4x)=3(5x+3)
Alright, team, let's get up close and personal with our challenge for the day: 2(3-4x)=3(5x+3). When you first look at an equation like this, it might seem a bit intimidating, right? All those numbers, letters, and parentheses – where do you even begin? But don't sweat it! Let's break down what each part means so we can understand what we're truly trying to accomplish. On the left side, we have 2(3-4x). This means we're multiplying the number 2 by everything inside the parentheses, which is 3-4x. The x here is our variable – it's the unknown value we're trying to figure out. The 3 is a constant, and the -4 is the coefficient of x. Pretty straightforward, right? Similarly, on the right side, we've got 3(5x+3). Here, the number 3 is multiplying everything inside its parentheses, which consists of 5x (where 5 is the coefficient of x) and another constant, 3. The entire goal of solving a linear equation like this is to find the single numerical value for x that makes the equation true. In other words, when you substitute that value of x back into the equation, the expression on the left side (LHS) must equal the expression on the right side (RHS). It's like finding the perfect balance point! This initial understanding of what each piece represents – constants, variables, coefficients, and operations (like multiplication implied by the parentheses) – is absolutely critical before you even think about moving numbers around. Many common mistakes happen when folks rush past this conceptual understanding. We're essentially looking for the magic number 'x' that satisfies this specific relationship. Our strategy will involve simplifying both sides, collecting like terms, and then isolating 'x' through a series of inverse operations. By carefully analyzing the structure of 2(3-4x)=3(5x+3), we prepare ourselves for the battle ahead, knowing exactly what tools we'll need and what our ultimate target is: that elusive value of 'x'.
Step-by-Step Solution: Conquering the Equation Like a Pro!
Now for the main event, guys – the actual solving process for 2(3-4x)=3(5x+3)! We're going to take this equation apart, piece by piece, ensuring every single move is logical and correct. No shortcuts, just pure, unadulterated algebra goodness. This section is all about building your confidence and showing you that even complex-looking equations can be tackled with a systematic approach. Remember, math is like a puzzle, and each step brings us closer to the final picture.
Step 1: Distribute and Simplify – Get Rid of Those Parentheses!
Our very first move in solving 2(3-4x)=3(5x+3) is to eliminate those parentheses. And for that, we bring in our good friend, the distributive property! This property tells us that to multiply a number by a sum or difference inside parentheses, you multiply that number by each term inside the parentheses. So, let's apply it to both sides:
-
Left Side (LHS): We have
2(3-4x). We'll multiply2by3and then2by-4x.2 * 3 = 62 * (-4x) = -8xSo, the left side becomes:6 - 8x -
Right Side (RHS): We have
3(5x+3). We'll multiply3by5xand then3by3.3 * 5x = 15x3 * 3 = 9So, the right side becomes:15x + 9
After applying the distributive property, our equation now looks much cleaner: 6 - 8x = 15x + 9. See? Already less intimidating! This step is absolutely fundamental because it transforms the equation from a parenthetical nightmare into a straightforward linear form that's much easier to manipulate. Always make distributing your first order of business when you see those parentheses. Skipping this or doing it incorrectly is a common pitfall, so take your time and double-check your multiplication and signs. A misplaced minus sign can throw off your entire solution!
Step 2: Gather Your X's – Bringing Like Terms Together!
Now that we have 6 - 8x = 15x + 9, our next goal is to get all the x terms on one side of the equation and all the constant terms (plain numbers) on the other. This helps us consolidate our unknown variable. It doesn't really matter which side you choose for x, but it's often a good strategy to move the x terms so that the coefficient of x remains positive, if possible, to avoid dealing with extra negative signs. In this case, we have -8x on the left and 15x on the right. To move -8x to the right side, we perform the inverse operation: we add 8x to both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep the equation balanced!
6 - 8x + 8x = 15x + 9 + 8x
This simplifies to:
6 = 23x + 9
Awesome! All our x terms are now happily residing on the right side. Next, we need to get the constants to the left. We have +9 on the right, so to move it to the left, we'll subtract 9 from both sides:
6 - 9 = 23x + 9 - 9
This leaves us with:
-3 = 23x
Boom! We've successfully gathered our terms. The equation is now super streamlined, with all the x terms consolidated and all the constants grouped together. This symmetrical arrangement is exactly what we need for the final push. Understanding how to isolate terms using inverse operations is a cornerstone of algebra, allowing us to systematically simplify equations. It's like separating the different colors of LEGO bricks before you start building your masterpiece.
Step 3: Isolate X – The Moment of Truth!
We're almost there, folks! Our equation is now -3 = 23x. Our final task is to get x all by itself. Right now, x is being multiplied by 23. To undo multiplication, we use its inverse operation: division. So, we need to divide both sides of the equation by 23.
-3 / 23 = 23x / 23
And voilà! This simplifies to:
x = -3/23
And there you have it! The value of x is -3/23. It's perfectly fine to get a fraction as an answer, so don't let that throw you off. In fact, in many real-world scenarios, answers aren't always neat whole numbers. This final step truly isolates the variable, providing the definitive solution to our linear equation. This entire sequence of distributive property, gathering like terms, and then isolating the variable through division or multiplication forms the backbone of solving virtually any linear equation. You've just conquered a significant algebraic challenge! Always remember the fundamental rule of keeping the equation balanced: whatever operation you perform on one side, you must perform on the other. This ensures that the equality holds true throughout the solving process and leads you to the correct value of 'x'.
Don't Just Solve It, Prove It! Checking Your Answer Is Super Important!
Alright, you've done the hard work, you've found x = -3/23. But how do you know if you're actually right? This, my friends, is where the