Hey guys! Let's dive into the world of algebra and tackle the equation 9(x + 7) + 3 = -2(x - 5) - 5. Solving for x might seem daunting at first, but with a step-by-step approach and a bit of algebraic maneuvering, we can conquer this equation and emerge victorious! In this article, we'll break down the process, making it super easy to follow. We will delve deep into the mechanics of solving linear equations, ensuring that you not only understand the solution to this specific problem but also gain a solid foundation for tackling similar algebraic challenges in the future. So, buckle up and get ready to become an x-solving pro!
Understanding the Equation
Before we jump into the nitty-gritty, let's take a good look at our equation: 9(x + 7) + 3 = -2(x - 5) - 5. The first thing you'll notice is that it's a linear equation, meaning the highest power of x is 1. This is great news because linear equations have a straightforward solving process. We've got parentheses, addition, subtraction, and multiplication all mixed in, but don't worry, we'll handle each operation one at a time. The key here is to remember the order of operations (PEMDAS/BODMAS) and apply it diligently. We need to simplify both sides of the equation by expanding any brackets and combining similar terms before we begin isolating our variable x. This initial simplification is crucial because it sets the stage for a smoother solving process, reducing the complexity of the equation and making it more manageable. Without this initial step, we might find ourselves lost in a maze of numbers and symbols, but with a clear strategy, we can make this process straightforward and enjoyable.
Step 1: Distribute
The first order of business is to get rid of those parentheses. We do this by distributing the numbers outside the parentheses to the terms inside. On the left side, we have 9(x + 7), which means we multiply 9 by both x and 7. This gives us 9 * x + 9 * 7 = 9x + 63. Remember, distribution is all about ensuring that each term inside the parentheses is correctly multiplied by the term outside. On the right side, we have -2(x - 5). Here, we need to be careful with the negative sign. We multiply -2 by both x and -5. This gives us -2 * x + (-2 * -5) = -2x + 10. Now, our equation looks like this: 9x + 63 + 3 = -2x + 10 - 5. Distributive property is a foundational concept in algebra, and mastering it allows us to handle complex equations with greater confidence. It is a fundamental tool that enables us to transform complex expressions into simpler, more manageable forms, paving the way for a clearer solution path. By meticulously applying the distributive property, we can avoid common pitfalls and ensure accuracy in our algebraic manipulations.
Step 2: Combine Like Terms
Now that we've distributed, let's simplify both sides of the equation by combining like terms. On the left side, we have 63 + 3, which adds up to 66. So, the left side becomes 9x + 66. On the right side, we have 10 - 5, which simplifies to 5. Thus, the right side becomes -2x + 5. Our equation now looks much cleaner: 9x + 66 = -2x + 5. Combining like terms is like tidying up a room; it makes everything more organized and easier to handle. By consolidating constant terms and grouping variable terms, we reduce the visual clutter and make the underlying structure of the equation more apparent. This simplification is not merely cosmetic; it directly contributes to our ability to isolate the variable and find its value. A well-organized equation is a half-solved equation, as it allows us to focus our efforts on the core task of variable isolation without being distracted by unnecessary complexity.
Isolating x
Now comes the fun part: isolating x. Our goal is to get all the x terms on one side of the equation and all the constant terms on the other side. This is achieved through a series of strategic moves, adding or subtracting terms from both sides to maintain the balance of the equation. Think of it like a seesaw: whatever you do to one side, you must do to the other to keep it level. The beauty of algebra lies in this principle of equality, which allows us to manipulate equations while preserving their fundamental truth. Each step we take is a calculated move, bringing us closer to the ultimate goal of revealing the value of x. It's a bit like a puzzle, where each correct move unveils a new piece of the solution, building towards the final, satisfying answer. So, let's continue our journey, carefully and deliberately, towards solving for x.
Step 3: Move x Terms to One Side
Let's move the x terms to the left side. We can do this by adding 2x to both sides of the equation. This cancels out the -2x on the right side: 9x + 66 + 2x = -2x + 5 + 2x. This simplifies to 11x + 66 = 5. Adding the same term to both sides is a fundamental operation in algebra, ensuring that the equation remains balanced while we strategically relocate terms. By adding 2x to both sides, we effectively neutralized the x term on the right-hand side, paving the way for isolating the variable on the left. This maneuver demonstrates the power of inverse operations in simplifying equations. We are essentially using addition to undo subtraction, a common tactic in algebraic problem-solving. This strategic addition not only simplifies the equation but also brings us one step closer to our ultimate goal of finding the value of x.
Step 4: Move Constant Terms to the Other Side
Now, let's move the constant terms to the right side. We can subtract 66 from both sides of the equation to cancel out the +66 on the left: 11x + 66 - 66 = 5 - 66. This simplifies to 11x = -61. Just as we added to move the x terms, we now subtract to relocate the constant terms. This consistent application of inverse operations is the cornerstone of algebraic manipulation. By subtracting 66 from both sides, we isolated the term containing x on the left, leaving us with a simple equation that directly relates x to a numerical value. This process underscores the elegance of algebraic transformations, where each operation is carefully chosen to bring us closer to the solution. The equation 11x = -61 is a significant milestone in our solving journey, as it represents a clear and concise relationship between x and a constant, setting the stage for the final step of division.
Step 5: Solve for x
Finally, to solve for x, we need to get x by itself. Since x is being multiplied by 11, we can undo this by dividing both sides of the equation by 11: 11x / 11 = -61 / 11. This gives us x = -61/11. And there you have it! We've solved for x. Division is the final act in our algebraic drama, the operation that reveals the true value of our sought-after variable. By dividing both sides of the equation by the coefficient of x, we isolate x and obtain the solution. This step is the culmination of all our previous efforts, the moment where the puzzle pieces finally fall into place. The result, x = -61/11, is not just a number; it's the answer to our algebraic quest, the solution that satisfies the original equation. It represents the value of x that makes the equation true, a testament to the power and precision of algebraic methods.
Final Answer
Therefore, the solution to the equation 9(x + 7) + 3 = -2(x - 5) - 5 is x = -61/11. Whew! We made it. Solving equations might seem tricky at first, but with practice and a clear understanding of the steps involved, you'll become a pro in no time. Remember, the key is to break down the problem into smaller, manageable steps and to apply the rules of algebra consistently. You've now seen how to tackle this equation, and you can apply these same principles to solve countless other algebraic challenges. So, keep practicing, keep exploring, and keep enjoying the world of mathematics! Algebraic equations, though sometimes daunting, are ultimately puzzles waiting to be solved. Each equation is a new challenge, an opportunity to apply your skills and knowledge to unravel a mathematical mystery. With each equation you solve, you not only expand your understanding of algebra but also sharpen your problem-solving abilities, skills that are valuable in many areas of life. So, embrace the challenge, dive into the world of equations, and watch your mathematical confidence soar.