Solving For X In The Equation 0x - 4.5 + 3x = 12x - 1.1

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Finding the value of 'x' that satisfies a given equation is a fundamental concept in algebra. This article delves into the step-by-step solution of the equation 0x - 4.5 + 3x = 12x - 1.1, offering a clear and comprehensive understanding of the algebraic manipulations involved. Whether you're a student grappling with algebraic equations or simply seeking to refresh your mathematical skills, this guide provides a detailed walkthrough to help you master the process. We will break down each step, explaining the rationale behind it, and ultimately arrive at the correct solution for 'x'. So, let's embark on this algebraic journey and unravel the mystery of this equation.

Understanding the Equation

Before diving into the solution, let's first understand the given equation: 0x - 4.5 + 3x = 12x - 1.1. This is a linear equation in one variable, 'x'. The goal is to isolate 'x' on one side of the equation to determine its value. To achieve this, we will use the properties of equality, which allow us to perform the same operations on both sides of the equation without changing its balance. These operations include addition, subtraction, multiplication, and division. The presence of the term 0x might seem unusual, but it simplifies to 0, which is a crucial first step in solving the equation. By carefully applying these principles, we can systematically solve for 'x' and find the value that makes the equation true. Let's begin the process of simplification and isolation to reveal the solution.

Step-by-Step Solution

1. Simplify the Equation

The first step in solving the equation 0x - 4.5 + 3x = 12x - 1.1 is to simplify both sides. We begin by recognizing that 0x equals 0, as any number multiplied by zero is zero. This simplifies the left side of the equation. Next, we combine like terms, specifically the terms containing 'x', to further simplify the equation. This process will make the equation more manageable and easier to work with. By reducing the number of terms on each side, we pave the way for isolating 'x'. This initial simplification is a critical step in solving any algebraic equation, as it sets the stage for subsequent operations. Let's proceed with this simplification to reveal the underlying structure of the equation.

  • Original Equation: 0x - 4.5 + 3x = 12x - 1.1
  • Simplify 0x: 0 - 4.5 + 3x = 12x - 1.1
  • Combine like terms: -4.5 + 3x = 12x - 1.1

2. Isolate x Terms

To isolate the 'x' terms, we need to move all terms containing 'x' to one side of the equation. This is achieved by performing the same operation on both sides, ensuring the equation remains balanced. In this case, we can subtract 3x from both sides of the equation -4.5 + 3x = 12x - 1.1. This will eliminate the 3x term from the left side and consolidate the 'x' terms on the right side. By strategically moving terms, we bring the equation closer to the form where 'x' can be easily isolated. This step is crucial in the process of solving for 'x', as it groups the variable terms together. Let's execute this step to further simplify the equation and pave the way for the final solution.

  • Equation: -4.5 + 3x = 12x - 1.1
  • Subtract 3x from both sides: -4.5 + 3x - 3x = 12x - 1.1 - 3x
  • Simplify: -4.5 = 9x - 1.1

3. Isolate Constant Terms

Next, we need to isolate the constant terms, which are the terms without any variables. To do this, we move all constant terms to the side of the equation that does not contain the 'x' term. In the equation -4.5 = 9x - 1.1, we can add 1.1 to both sides. This will eliminate the -1.1 from the right side and consolidate the constant terms on the left side. This step is essential in isolating 'x', as it separates the variable term from the constant terms. By performing this operation, we are one step closer to solving for 'x'. Let's proceed with this step to further simplify the equation.

  • Equation: -4.5 = 9x - 1.1
  • Add 1.1 to both sides: -4.5 + 1.1 = 9x - 1.1 + 1.1
  • Simplify: -3.4 = 9x

4. Solve for x

Now that we have isolated the 'x' term on one side and the constant terms on the other, we can solve for 'x'. The equation is currently in the form -3.4 = 9x. To find the value of 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 9. This will isolate 'x' and give us its value. This final step is the culmination of all the previous steps, and it reveals the solution to the equation. By performing this division, we will determine the value of 'x' that satisfies the original equation. Let's complete this step and unveil the solution.

  • Equation: -3.4 = 9x
  • Divide both sides by 9: -3.4 / 9 = 9x / 9
  • Simplify: x = -3.4 / 9
  • Calculate: x = -0.3777...
  • Round to the nearest tenth: x ≈ -0.38

Final Answer

After performing all the algebraic manipulations, we have arrived at the solution for 'x'. The value of 'x' that satisfies the equation 0x - 4.5 + 3x = 12x - 1.1 is approximately -0.38. This solution is obtained by simplifying the equation, isolating the 'x' terms, isolating the constant terms, and finally dividing by the coefficient of 'x'. This step-by-step process demonstrates how algebraic equations can be systematically solved using the properties of equality. Understanding these principles is crucial for mastering algebra and solving more complex equations. Therefore, the final answer to the equation is:

x ≈ -0.38

Conclusion

In conclusion, solving the equation 0x - 4.5 + 3x = 12x - 1.1 involves a series of algebraic manipulations aimed at isolating the variable 'x'. We began by simplifying the equation, then systematically moved terms to isolate 'x' on one side. This process included combining like terms, subtracting 'x' terms from both sides, and adding constant terms to both sides. Finally, we divided by the coefficient of 'x' to find its value. The solution we obtained is approximately -0.38. This exercise highlights the importance of understanding the properties of equality and applying them methodically to solve equations. Mastering these techniques is essential for success in algebra and beyond. By practicing and understanding these steps, you can confidently tackle various algebraic equations and find their solutions. Remember, the key to success in mathematics is consistent practice and a clear understanding of the underlying principles.