Solving -9(x+3)+12=-3(2x+5)-3x Determining The Solution Type

In this article, we will delve into the process of solving the given equation and identifying the nature of its solutions. The equation we are tasked with analyzing is: $-9(x+3)+12=-3(2x+5)-3x$. This equation is a linear equation, and our goal is to determine whether it has a unique solution, no solution, or infinitely many solutions. Understanding the properties of linear equations is crucial for success in algebra and various mathematical applications. To accurately solve the equation, we will methodically simplify both sides, combine like terms, and isolate the variable x. This step-by-step approach will help us avoid common algebraic errors and arrive at the correct conclusion. The ability to solve equations forms the foundation for more advanced mathematical concepts and real-world problem-solving scenarios. By the end of this article, you will have a clear understanding of how to tackle such equations and confidently determine the nature of their solutions.

Step-by-Step Solution

1. Distribute and Simplify

First, let's distribute the constants on both sides of the equation to eliminate the parentheses. On the left side, we distribute -9 across (x + 3), and on the right side, we distribute -3 across (2x + 5). This gives us:

$-9(x+3)+12 = -9x - 27 + 12$

$-3(2x+5)-3x = -6x - 15 - 3x$

Now, we simplify each side by combining like terms. On the left side, we combine -27 and 12, and on the right side, we combine -6x and -3x:

$-9x - 15 = -9x - 15$

This simplified form makes it easier to see the relationship between the two sides of the equation. Recognizing the importance of simplification in solving equations can save time and reduce errors. Simplifying both sides of an equation allows us to clearly see the underlying structure and determine the next steps. The distribution and simplification steps are foundational techniques in algebraic manipulation, and proficiency in these techniques is essential for solving a wide range of mathematical problems. By carefully applying the distributive property and combining like terms, we ensure that the equation remains balanced and that we are progressing towards the correct solution. The simplicity of the resulting equation suggests a special case, which we will explore further in the next steps.

2. Analyze the Simplified Equation

After simplifying, we have the equation: $-9x - 15 = -9x - 15$. Notice that both sides of the equation are exactly the same. This means that no matter what value we substitute for x, the equation will always hold true. This is a key indicator of an equation having infinitely many solutions. Understanding the implications of identical expressions on both sides of an equation is crucial for correctly interpreting the solution set. When the left and right sides of an equation are identical, it signifies that any real number value for the variable will satisfy the equation. This situation arises because the equation essentially represents an identity, a mathematical statement that is always true. The presence of infinitely many solutions implies that the variable x is not restricted to a specific value; instead, it can take on any value within the set of real numbers. This concept is fundamental in linear algebra and differential equations, where identifying the nature of solution sets is critical. Recognizing such cases early in the problem-solving process can save time and effort, preventing unnecessary attempts to find a unique solution. The equation's structure, showing identical expressions on both sides, provides a clear indication of its solution characteristics.

3. Determine the Number of Solutions

Since both sides of the equation are identical, this equation has infinitely many solutions. This is because any value of x will satisfy the equation. For instance, if we add 9x to both sides, we get -15 = -15, which is always true. Similarly, if we add 15 to both sides, we obtain -9x = -9x, which also holds true for any x. The concept of infinitely many solutions arises when an equation simplifies to an identity, a statement that is always true regardless of the variable's value. Such identities indicate that the equation represents a fundamental relationship that is universally valid within the domain of real numbers. The ability to recognize equations with infinitely many solutions is essential for understanding linear systems and their behavior. In practical applications, equations with infinitely many solutions may represent scenarios where there are multiple valid solutions to a problem, or where the problem's constraints do not uniquely determine a single solution. This type of solution set differs significantly from equations with a single, unique solution or equations with no solution, each of which has its own implications and interpretations. Understanding these distinctions is vital for effective problem-solving in mathematics and related fields.

Choosing the Correct Answer

Based on our step-by-step solution, we have determined that the equation $-9(x+3)+12=-3(2x+5)-3x$ has infinitely many solutions. Therefore, the correct answer is:

D. The equation has infinitely many solutions.

Selecting the correct answer requires a thorough understanding of the solution process and the nature of different types of solutions. In this case, the simplification and analysis of the equation led us to the conclusion that any value of x would satisfy the equation, indicating an infinite number of solutions. This contrasts with equations that have a unique solution, where only one value of the variable makes the equation true, or equations with no solution, where no value of the variable can satisfy the equation. The ability to distinguish between these solution types is a fundamental skill in algebra and is crucial for success in more advanced mathematical studies. When facing multiple-choice questions, it is essential to carefully evaluate each option in light of the derived solution. Misinterpreting the results or overlooking key steps in the solving process can lead to selecting an incorrect answer. The process of elimination can also be a valuable strategy, where incorrect options are ruled out based on the understanding of the problem, thereby increasing the chances of choosing the correct response. In this particular instance, the equation's inherent structure and its simplification pathway clearly pointed towards the existence of infinitely many solutions.

Conclusion

In summary, we solved the equation $-9(x+3)+12=-3(2x+5)-3x$ and determined that it has infinitely many solutions. This understanding is crucial for solving more complex algebraic problems. The process of solving equations involves simplification, distribution, combining like terms, and analyzing the resulting expression to determine the nature of the solutions. In this case, the equation simplified to an identity, where both sides were identical, indicating that any value of x would satisfy the equation. The ability to recognize and handle equations with infinitely many solutions is a valuable skill in mathematics, with applications in various fields such as physics, engineering, and computer science. Understanding the underlying principles and techniques allows for more efficient problem-solving and a deeper appreciation of mathematical concepts. Furthermore, mastering the steps involved in solving linear equations provides a strong foundation for tackling more advanced topics, including systems of equations, inequalities, and higher-degree polynomials. The methodical approach used in this article—distributing, simplifying, and analyzing—can be applied to a wide range of mathematical problems, making it a versatile and essential skill for any student or professional dealing with quantitative analysis.