Equation Solution Analysis -9(x+3) + 12 = -3(2x+5) - 3x

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In the realm of mathematics, linear equations form the bedrock of algebraic problem-solving. These equations, characterized by a variable raised to the first power, are ubiquitous in various fields, from physics and engineering to economics and computer science. This article delves into the intricacies of solving a specific linear equation, -9(x+3)+12=-3(2x+5)-3x, providing a comprehensive, step-by-step solution and clarifying the underlying principles involved. Our primary goal is to determine the nature of the solution – whether it has a unique solution, no solution, or infinitely many solutions. Understanding these concepts is crucial for mastering algebra and its applications.

The Given Equation: A Closer Look

Let's begin by revisiting the equation at hand: -9(x+3)+12=-3(2x+5)-3x. This equation presents a typical linear equation structure, featuring the variable 'x' on both sides. To solve it effectively, we must strategically employ algebraic manipulations to isolate 'x' on one side of the equation. This involves applying the distributive property, combining like terms, and carefully balancing the equation to maintain equality. By meticulously following these steps, we can unravel the solution and determine its nature.

Step 1: Applying the Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions involving parentheses. In our equation, we have two instances where the distributive property can be applied: on the left side with -9(x+3) and on the right side with -3(2x+5). Let's apply this property to both sides:

  • On the left side: -9(x+3) expands to -9 * x + (-9) * 3, which simplifies to -9x - 27.
  • On the right side: -3(2x+5) expands to -3 * 2x + (-3) * 5, which simplifies to -6x - 15.

Applying these expansions, our equation now becomes: -9x - 27 + 12 = -6x - 15 - 3x. This step is crucial as it eliminates the parentheses, paving the way for further simplification.

Step 2: Combining Like Terms

Now that we've applied the distributive property, the next step is to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power. In our equation, we have constant terms (-27 and 12 on the left, -15 on the right) and 'x' terms (-9x on the left, -6x and -3x on the right). Let's combine them:

  • On the left side: -27 + 12 simplifies to -15. So, the left side becomes -9x - 15.
  • On the right side: -6x - 3x simplifies to -9x. So, the right side becomes -9x - 15.

After combining like terms, our equation is now: -9x - 15 = -9x - 15. This simplified form reveals a crucial characteristic of the equation.

Step 3: Analyzing the Simplified Equation

Observe the equation -9x - 15 = -9x - 15. What do you notice? 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 hallmark of an equation with infinitely many solutions.

To further illustrate this, let's try to isolate 'x' as we would in a typical equation-solving scenario. Add 9x to both sides of the equation:

  • -9x - 15 + 9x = -9x - 15 + 9x
  • This simplifies to -15 = -15

The variable 'x' has completely disappeared, and we are left with a true statement: -15 is indeed equal to -15. This confirms our earlier observation that the equation holds true for any value of 'x'.

Determining the Nature of the Solution

Based on our step-by-step simplification and analysis, we can definitively conclude that the equation -9(x+3)+12=-3(2x+5)-3x has infinitely many solutions. This is because the equation simplifies to an identity, a statement that is always true regardless of the value of the variable. When an equation reduces to an identity, it signifies that the two sides of the equation represent the same line, and thus, every point on the line is a solution.

Understanding Infinite Solutions

Equations with infinitely many solutions are not uncommon in algebra. They arise when the coefficients and constants on both sides of the equation are proportional. In our case, the coefficients of 'x' (-9) and the constant terms (-15) are identical on both sides, leading to the identity -15 = -15. This concept is crucial in understanding the behavior of linear equations and their graphical representations.

Conclusion: The Equation's Infinite Embrace

In summary, we have meticulously solved the equation -9(x+3)+12=-3(2x+5)-3x, employing the distributive property, combining like terms, and analyzing the resulting simplified equation. Our analysis reveals that the equation has infinitely many solutions. This outcome underscores the importance of careful algebraic manipulation and the ability to recognize identities. Understanding the concept of infinitely many solutions is fundamental to mastering linear equations and their diverse applications in mathematics and beyond. By grasping these principles, you'll be well-equipped to tackle a wide range of algebraic problems with confidence and precision.

Which of the following statements accurately describes the solution to the equation -9(x+3) + 12 = -3(2x+5) - 3x?

A. The equation has one solution, x = 1.

B. The equation has one solution, x = 0.

C. The equation has no solution.

D. The equation has infinitely many solutions.