Solving -9(x+3)+12=-3(2x+5)-3x A Step-by-Step Guide

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In this article, we will embark on a journey to dissect and solve the mathematical equation -9(x+3)+12=-3(2x+5)-3x. This equation, seemingly complex at first glance, holds the key to understanding fundamental algebraic principles. Our mission is to unravel its intricacies, identify the correct solution, and explore the underlying concepts that govern its behavior. We will meticulously analyze each step, providing a clear and concise explanation to ensure a comprehensive understanding of the solution process. Whether you are a student seeking to master algebraic equations or simply a curious mind eager to explore the world of mathematics, this article will serve as your guide. We will delve into the properties of equality, the distributive property, and the art of simplifying expressions. By the end of this exploration, you will not only be able to solve this specific equation but also possess a deeper appreciation for the elegance and power of algebraic manipulation.

Understanding the Foundation: The Distributive Property

Before we dive into the nitty-gritty of solving the equation, it is crucial to grasp the foundational concept of the distributive property. This property is the cornerstone of simplifying expressions that involve parentheses. In essence, the distributive property states that multiplying a number by a sum or difference inside parentheses is the same as multiplying the number by each term inside the parentheses individually and then adding or subtracting the results. Mathematically, this can be expressed as: a(b + c) = ab + ac. This seemingly simple rule is a powerful tool in algebra, allowing us to break down complex expressions into more manageable parts. In our equation, -9(x+3)+12=-3(2x+5)-3x, the distributive property will be our first line of attack. We will apply it to both sides of the equation, carefully multiplying the constants outside the parentheses by each term inside. This initial step will pave the way for further simplification and ultimately lead us to the solution. Understanding the distributive property is not just about memorizing a formula; it's about grasping the underlying logic that allows us to manipulate expressions and reveal their hidden structure. This understanding will empower you to tackle a wide range of algebraic problems with confidence and precision.

Step-by-Step Solution: Unraveling the Equation

Now, let's embark on the step-by-step journey of solving the equation -9(x+3)+12=-3(2x+5)-3x. Our first objective is to eliminate the parentheses by applying the distributive property. On the left side of the equation, we have -9(x+3). Distributing the -9, we get -9 * x + (-9) * 3, which simplifies to -9x - 27. Adding the +12 from the original equation, the left side now reads -9x - 27 + 12. On the right side, we have -3(2x+5). Distributing the -3, we get -3 * 2x + (-3) * 5, which simplifies to -6x - 15. Adding the -3x from the original equation, the right side becomes -6x - 15 - 3x. Now our equation looks like this: -9x - 27 + 12 = -6x - 15 - 3x. The next step is to simplify both sides of the equation by combining like terms. On the left side, we have -27 and +12, which combine to -15. So the left side simplifies to -9x - 15. On the right side, we have -6x and -3x, which combine to -9x. So the right side simplifies to -9x - 15. Our equation now reads -9x - 15 = -9x - 15. At this point, we can see that both sides of the equation are identical. This indicates a special type of solution. To further confirm, let's add 9x to both sides of the equation. This gives us -15 = -15, which is a true statement. This true statement, independent of the value of x, reveals that the equation has infinitely many solutions. Any value of x will satisfy the equation.

Identifying the Correct Solution: Analyzing the Options

Having meticulously solved the equation -9(x+3)+12=-3(2x+5)-3x, we arrive at the crucial step of identifying the correct solution from the given options. We recall that our step-by-step simplification led us to the equation -9x - 15 = -9x - 15, which further simplified to -15 = -15. This final statement, being a universally true identity, signifies that the equation possesses infinitely many solutions. Now, let's examine the options provided: A. The equation has one solution, x=1. This is incorrect, as we've established that the equation has infinitely many solutions, not just one. B. The equation has one solution, x=0. This is also incorrect for the same reason as option A. C. The equation has no solution. This is incorrect because our simplification process led to a true statement, indicating the existence of solutions. D. The equation has infinitely many solutions. This is the correct answer. Our detailed analysis and simplification process definitively demonstrate that any value of x will satisfy the original equation. Therefore, the equation has an infinite number of solutions. The process of elimination further solidifies our conclusion, as the other options are demonstrably false. This rigorous approach ensures that we not only arrive at the correct answer but also understand the reasoning behind it.

Delving Deeper: Understanding Infinite Solutions

The concept of an equation having infinitely many solutions can sometimes seem perplexing. To truly grasp this idea, let's delve deeper into the underlying principles. An equation with infinitely many solutions, often referred to as an identity, is an equation that holds true for any value of the variable. In our case, the equation -9(x+3)+12=-3(2x+5)-3x simplifies to -15 = -15. This statement is independent of x; it is true regardless of what value we substitute for x. This is the hallmark of an identity. To visualize this, imagine plotting the graphs of the expressions on both sides of the original equation. If the equation has infinitely many solutions, the graphs will coincide; they will be the same line. This means that for any x-value you choose, the corresponding y-values on both graphs will be equal. Conversely, an equation with no solution will have graphs that are parallel lines, never intersecting. An equation with one solution will have graphs that intersect at a single point. Understanding the graphical representation of equations provides a powerful visual aid for comprehending the nature of their solutions. In the case of infinite solutions, the equation essentially represents the same mathematical relationship expressed in two different forms. This deeper understanding allows us to move beyond rote memorization and appreciate the interconnectedness of mathematical concepts.

Avoiding Common Pitfalls: A Guide to Accuracy

When solving algebraic equations, it's easy to make small errors that can lead to incorrect solutions. To ensure accuracy, it's crucial to be mindful of common pitfalls and adopt strategies to avoid them. One common mistake is incorrectly applying the distributive property. Remember to multiply the term outside the parentheses by every term inside the parentheses, paying close attention to signs. Another pitfall is combining unlike terms. Only terms with the same variable and exponent can be combined. For example, -9x and -6x can be combined, but -9x and -15 cannot. Sign errors are also a frequent source of mistakes. When moving terms across the equals sign, remember to change their signs. Additionally, it's essential to perform operations in the correct order (PEMDAS/BODMAS). Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). To minimize errors, it's helpful to write out each step clearly and neatly. This allows you to track your work and easily identify any mistakes. Furthermore, checking your solution by substituting it back into the original equation is a valuable practice. If the equation holds true, your solution is correct. By being aware of these common pitfalls and adopting careful working habits, you can significantly improve your accuracy and confidence in solving algebraic equations.

Conclusion: Mastering Algebraic Equations

In conclusion, we have successfully navigated the process of solving the equation -9(x+3)+12=-3(2x+5)-3x, demonstrating that it possesses infinitely many solutions. This journey has reinforced our understanding of fundamental algebraic concepts such as the distributive property, combining like terms, and the nature of equations with infinite solutions. We have also highlighted common pitfalls to avoid and strategies for ensuring accuracy. Mastering algebraic equations is not merely about finding the right answer; it's about developing a deep understanding of the underlying principles and cultivating problem-solving skills. The ability to manipulate equations, simplify expressions, and interpret solutions is a valuable asset in various fields, from mathematics and science to engineering and finance. By embracing a methodical approach, paying attention to detail, and continuously practicing, you can unlock the power of algebra and confidently tackle complex mathematical challenges. Remember that mathematics is not a spectator sport; it's an active engagement. The more you practice, the more fluent you will become in the language of numbers and symbols. So, continue to explore, challenge yourself, and embrace the beauty and power of mathematics.

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