Equivalent Equations Solving For X In (5/6)x + (2/3) = -9

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In the realm of mathematics, solving equations is a fundamental skill. It allows us to find the value of an unknown variable, often represented by x. One common type of equation is a linear equation, where the variable is raised to the power of 1. In this article, we will delve into the process of solving a linear equation and, more importantly, identifying other equations that share the same solution. Our journey begins with the equation (5/6)x + (2/3) = -9. The core challenge is to find equivalent equations, those that, despite their different appearances, lead to the identical value for x. The process involves applying algebraic manipulations to both sides of the equation, maintaining the balance while simplifying and isolating the variable. Each step taken must adhere to the fundamental principles of equality: whatever operation is performed on one side must be mirrored on the other. This exploration is not just about finding the solution; it's about understanding the underlying structure of equations and the transformations that preserve their solutions. This concept is pivotal in higher mathematics, where the ability to manipulate equations and recognize equivalent forms is crucial for problem-solving and proving theorems. Moreover, it highlights the power of algebra as a tool for representing and solving real-world problems, where different formulations of a problem can lead to the same solution, providing flexibility and insight in the modeling process.

Unraveling the Initial Equation: (5/6)x + (2/3) = -9

Let's start with our initial equation: (5/6)x + (2/3) = -9. To solve for x, we need to isolate it on one side of the equation. This involves a series of steps, each carefully chosen to simplify the equation while preserving its balance. The first step typically involves dealing with any fractions present. In our case, we have fractions with denominators 6 and 3. A common strategy is to multiply both sides of the equation by the least common multiple (LCM) of these denominators. The LCM of 6 and 3 is 6. By multiplying both sides by 6, we aim to eliminate the fractions, making the equation easier to work with. This is based on the fundamental property of equality, which states that multiplying both sides of an equation by the same non-zero number maintains the equality. The choice of the LCM is strategic because it ensures that both denominators will divide evenly into the multiplier, effectively canceling out the fractions. However, it's crucial to distribute the multiplication correctly on the left side, ensuring that each term is multiplied by 6. This step is a bridge from the fractional equation to a simpler, integer-based equation, a form that is often more intuitive and easier to manipulate. Furthermore, this approach highlights a common theme in mathematical problem-solving: transforming a complex problem into a simpler, equivalent one. The ability to recognize and apply such transformations is a hallmark of mathematical proficiency, allowing us to tackle seemingly daunting problems by breaking them down into manageable steps. This initial step sets the stage for further simplification and isolation of the variable, paving the way for the ultimate solution.

Transforming the Equation: A Step-by-Step Guide

Following our strategy, we multiply both sides of the equation (5/6)x + (2/3) = -9 by 6. This gives us 6 * [(5/6)x + (2/3)] = 6 * (-9). Now, we distribute the 6 on the left side: 6 * (5/6)x + 6 * (2/3) = -54. This simplifies to 5x + 4 = -54. This equation is a crucial step in our journey. It represents a transformation of the original equation into a simpler form, free of fractions. The process of distribution is a fundamental algebraic technique, ensuring that each term within the parentheses is correctly accounted for. The resulting equation, 5x + 4 = -54, is a linear equation in standard form, making it easier to isolate x. This form highlights the coefficient of x (which is 5) and the constant term (which is 4). The next step involves isolating the term with x on one side of the equation. This is achieved by subtracting 4 from both sides, maintaining the balance of the equation. This operation is based on the additive property of equality, which states that adding or subtracting the same value from both sides of an equation does not change its solution. The choice of subtracting 4 is strategic, as it cancels out the +4 on the left side, bringing us closer to isolating x. This step is a testament to the systematic nature of equation solving, where each operation is carefully chosen to achieve a specific goal. By consistently applying these principles, we can navigate through complex equations and arrive at the desired solution.

Isolating the Variable: Unveiling the Value of x

Continuing our quest to solve for x, we now have the equation 5x + 4 = -54. To isolate the term with x, we subtract 4 from both sides, resulting in 5x + 4 - 4 = -54 - 4, which simplifies to 5x = -58. This step is a direct application of the subtraction property of equality, ensuring that the equation remains balanced while moving us closer to the solution. Now, the equation is in a much simpler form, with only one term involving x on the left side. To finally isolate x, we need to undo the multiplication by 5. This is achieved by dividing both sides of the equation by 5. This is based on the division property of equality, which states that dividing both sides of an equation by the same non-zero number preserves the equality. Performing this division, we get (5x)/5 = (-58)/5, which simplifies to x = -58/5. This is the solution to the original equation. It represents the value of x that satisfies the equation (5/6)x + (2/3) = -9. The solution, -58/5, is a fraction, which is perfectly acceptable. It's important to remember that solutions to equations can be integers, fractions, or even irrational numbers. The key is to follow the algebraic principles correctly to arrive at the accurate solution. This final step completes the process of solving for x, demonstrating the power of algebraic manipulation in uncovering the value of an unknown variable.

Identifying Equivalent Equations: A Comparative Analysis

Now that we have found the solution for x, which is -58/5, the next crucial step is to identify which of the provided equations share the same solution. This involves comparing the given options with the steps we took to solve the original equation. Equivalent equations are those that, despite their different forms, ultimately lead to the same solution for x. To determine equivalence, we can either solve each equation individually and compare the solutions or, more efficiently, check if the equations can be derived from the original equation through valid algebraic manipulations. Let's analyze each option:

  1. 6(5/6 x + 2/3) = -9: This equation is obtained by multiplying both sides of the original equation by 6. This is a valid algebraic manipulation, based on the multiplication property of equality. Therefore, this equation is equivalent to the original.

  2. 6(5/6 x + 2/3) = -9(6): This equation also starts by multiplying both sides of the original equation by 6, but it goes a step further by multiplying the right side (-9) by 6 as well. This is also a valid manipulation, making this equation equivalent.

  3. 5x + 4 = -54: This equation is the result of simplifying 6(5/6 x + 2/3) = -9. It's a crucial step in solving the original equation, and hence, it is equivalent.

  4. 5x + 4 = -9: This equation is not equivalent. It seems to have missed the multiplication of -9 by 6 when clearing the fractions, leading to a different equation and, consequently, a different solution.

  5. 5x = -13: This equation is not equivalent. It doesn't follow logically from any valid manipulation of the original equation or its equivalent forms.

  6. 5x = -58: This equation is equivalent. It's the result of subtracting 4 from both sides of 5x + 4 = -54, a step in the correct solution process.

This comparative analysis highlights the importance of understanding the properties of equality and how they can be used to transform equations while preserving their solutions. By systematically analyzing each option, we can confidently identify the equations that share the same solution for x.

The Trio of Equivalent Equations: Our Final Selection

Based on our analysis, the three equations that have the same value of x as the original equation (5/6)x + (2/3) = -9 are:

  1. 6(5/6 x + 2/3) = -9: This equation is obtained by multiplying both sides of the original equation by 6, a valid algebraic manipulation.
  2. 6(5/6 x + 2/3) = -9(6): This equation also results from multiplying both sides of the original equation by 6 and explicitly showing the multiplication on the right side.
  3. 5x + 4 = -54: This equation is a simplified form of the original equation, obtained after multiplying by 6 and distributing.
  4. 5x = -58: This equation is obtained by subtracting 4 from both sides of 5x + 4 = -54.

These equations are all interconnected, representing different stages in the process of solving for x. They are equivalent because they all share the same solution, x = -58/5. This exercise underscores the concept of equivalent equations and how algebraic manipulations can transform the appearance of an equation without altering its solution. The ability to recognize and manipulate equivalent equations is a fundamental skill in algebra and is crucial for solving more complex problems. It also highlights the flexibility in representing mathematical relationships, where different forms can convey the same information.

Conclusion: The Power of Equivalent Equations

In conclusion, we've journeyed through the process of solving the linear equation (5/6)x + (2/3) = -9 and, more importantly, identified other equations that share the same solution. This exploration highlights the fundamental concept of equivalent equations in mathematics. Equivalent equations, while appearing different, possess the same solution set. This principle is a cornerstone of algebra, allowing us to manipulate equations into more convenient forms without altering their underlying meaning. The ability to recognize and generate equivalent equations is not just a matter of technique; it's a deep insight into the structure of mathematical relationships. By applying the properties of equality, we can confidently transform equations, simplifying them, and ultimately, solving for the unknown variable. This skill is invaluable in various mathematical contexts, from solving simple linear equations to tackling complex systems of equations and beyond. Moreover, the concept of equivalence extends beyond equations to other mathematical objects, such as fractions and expressions, making it a unifying theme in the discipline. The process of finding equivalent equations also reinforces the importance of precision and attention to detail in algebraic manipulations. A single error in applying the properties of equality can lead to a non-equivalent equation and an incorrect solution. Therefore, a thorough understanding of these principles is essential for mathematical accuracy and problem-solving success. Ultimately, the exploration of equivalent equations empowers us to approach mathematical challenges with greater flexibility and confidence, enabling us to navigate the complexities of algebra and beyond.