Equations With The Same Solution For X As (5/6)x + (2/3) = -9

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In the realm of mathematics, solving equations is a fundamental skill. Our focus here is to identify equations that share the same solution for x as the given equation: (5/6)x + (2/3) = -9. This involves manipulating the equation while preserving its solution set, primarily by applying algebraic principles such as the distributive property and the properties of equality. The goal is to find three equivalent equations from the given options. To master solving linear equations, it's crucial to understand the properties that allow us to manipulate equations without changing their solutions. These properties include the addition and multiplication properties of equality, which state that adding or multiplying both sides of an equation by the same number preserves the equality. Additionally, the distributive property, which allows us to expand expressions like a(b + c) into ab + ac, is essential for simplifying and solving equations. When tackling equations involving fractions, as in our case, clearing the fractions is often a helpful first step. This involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which eliminates the fractions and makes the equation easier to work with. However, it's essential to ensure that each term on both sides of the equation is multiplied by the LCM to maintain the balance of the equation. Furthermore, understanding the concept of inverse operations is key to isolating the variable x. Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division. By applying inverse operations to both sides of the equation, we can gradually isolate x and determine its value. In addition to mastering these fundamental principles, practice is crucial for developing proficiency in solving linear equations. The more equations you solve, the more comfortable you'll become with identifying the steps needed to isolate the variable and find the solution. Remember to always check your solution by substituting it back into the original equation to ensure that it satisfies the equation. This not only confirms the correctness of your solution but also helps reinforce your understanding of the equation-solving process. By mastering these skills, you'll be well-equipped to tackle a wide range of mathematical problems and applications.

Step-by-Step Solution

To determine which equations have the same value of x as the original equation, we will meticulously transform the original equation and compare it to the provided options.

  1. Original Equation: (5/6)x + (2/3) = -9

  2. Eliminate Fractions: To simplify the equation, we can eliminate fractions by multiplying both sides by the least common multiple (LCM) of the denominators, which is 6.

    • 6 * ((5/6)x + (2/3)) = 6 * (-9)
    • Applying the distributive property, we get:
      • 6 * (5/6)x + 6 * (2/3) = -54
      • This simplifies to:
        • 5x + 4 = -54

  3. Isolate the x term: Subtract 4 from both sides of the equation to isolate the term with x.

    • 5x + 4 - 4 = -54 - 4
    • This gives us:
      • 5x = -58

Analyzing the Options

Now, let's evaluate each of the provided options to see which ones match our transformed equations.

  • Option 1: 6((5/6)x + (2/3)) = -9

    • This equation is the result of multiplying only the left side of the original equation by 6, which alters the equation's balance and thus changes the solution. Therefore, this option is incorrect. Equations must be modified equally on both sides to maintain equivalence. When solving equations, the fundamental principle is to maintain balance. Any operation performed on one side must be mirrored on the other side to preserve the equality. In this case, multiplying only the left side of the equation by 6 disrupts this balance, leading to a different solution for x. The correct approach involves multiplying both sides of the equation by the same factor to eliminate fractions or simplify expressions while preserving the equation's integrity. This ensures that the solution remains consistent throughout the transformation process. Moreover, understanding the concept of equivalent equations is crucial. Equivalent equations are equations that have the same solution set. Manipulating an equation in a way that alters its solution set, such as multiplying only one side by a constant, creates a new equation that is not equivalent to the original. Therefore, it's essential to perform operations on both sides of the equation to maintain equivalence and ensure that the solution remains unchanged. This principle applies to all types of equations, from simple linear equations to more complex algebraic expressions. By adhering to this fundamental rule, we can confidently solve equations and find accurate solutions.

  • Option 2: 6((5/6)x + (2/3)) = -9(6)

    • This equation correctly multiplies both sides of the original equation by 6. Applying the distributive property on the left side yields 5x + 4, and the right side simplifies to -54. So, the equation becomes:

      • 5x + 4 = -54

    • This equation matches one of our transformed equations, making it a correct option. This step demonstrates a proper application of the multiplication property of equality. By multiplying both sides of the equation by 6, we eliminate the fractions and simplify the expression without altering the solution. This technique is commonly used when dealing with equations involving fractions or decimals, as it makes the equation easier to solve. Furthermore, this step highlights the importance of simplifying equations to isolate the variable. Once the fractions are eliminated, the equation becomes more manageable, and we can proceed with the next steps to isolate x. This often involves performing inverse operations, such as adding or subtracting constants from both sides of the equation. By following a systematic approach of simplification and isolation, we can efficiently solve for the variable and find the solution to the equation. Moreover, this step reinforces the concept of maintaining balance in equations. Just as in the previous step, the key is to perform the same operation on both sides of the equation to preserve the equality. This ensures that the solution remains consistent throughout the simplification process. By adhering to this principle, we can confidently manipulate equations and arrive at the correct solution.

  • Option 3: 5x + 4 = -54

    • As we derived in step 2, this equation is equivalent to the original equation after multiplying both sides by 6 and simplifying. Thus, this is a correct option. This transformation is a crucial step in solving the equation, as it eliminates the fractions and makes the equation easier to work with. By multiplying both sides by the least common multiple of the denominators, we clear the fractions and obtain an equation with integer coefficients. This simplifies the subsequent steps and allows us to apply algebraic techniques more easily. Furthermore, this step reinforces the importance of applying the distributive property correctly. When multiplying a sum or difference by a constant, we must distribute the constant to each term within the parentheses. In this case, multiplying 6 by (5/6)x + (2/3) requires distributing the 6 to both the (5/6)x term and the (2/3) term. Failing to apply the distributive property correctly can lead to errors in the solution. Moreover, this step highlights the power of simplification in solving equations. By simplifying the equation, we reduce its complexity and make it more manageable. This often involves combining like terms, eliminating fractions or decimals, and applying algebraic identities. Simplification not only makes the equation easier to solve but also reduces the chances of making errors along the way. By mastering simplification techniques, we can approach complex equations with confidence and find accurate solutions efficiently.

  • Option 4: 5x + 4 = -9

    • This equation is incorrect because it does not account for multiplying the right side of the original equation by 6 when clearing fractions. Therefore, the solution for x will be different. This highlights the critical importance of maintaining balance in equations. When performing operations on one side of the equation, we must perform the same operations on the other side to preserve the equality. In this case, multiplying the left side of the equation by 6 to eliminate fractions requires multiplying the right side by 6 as well. Failing to do so disrupts the balance and leads to an incorrect solution. Moreover, this step underscores the need for careful attention to detail when solving equations. Even a small error, such as neglecting to multiply a term by the correct factor, can significantly impact the solution. Therefore, it's essential to double-check each step and ensure that all operations are performed correctly. This includes paying attention to signs, coefficients, and the order of operations. Furthermore, this step reinforces the concept of equivalent equations. Equivalent equations are equations that have the same solution set. To transform an equation into an equivalent equation, we must perform operations that preserve the equality. Multiplying or dividing both sides of the equation by the same nonzero constant, adding or subtracting the same quantity from both sides, and simplifying expressions are all valid operations that maintain equivalence. However, performing operations that disrupt the balance of the equation, such as multiplying only one side by a constant, can lead to a nonequivalent equation with a different solution.

  • Option 5: 5x = -13

    • This equation is obtained by incorrectly subtracting 4 from -9 (5x + 4 = -9), which results in 5x = -13. However, the correct equation after subtracting 4 from both sides of 5x + 4 = -54 is 5x = -58. Therefore, this option is incorrect. This highlights the importance of accurately applying the properties of equality. When solving equations, we must perform operations correctly on both sides to maintain the balance of the equation. In this case, the error lies in incorrectly subtracting 4 from -9 instead of -54. This simple mistake leads to a completely different equation with a different solution. Moreover, this step underscores the need for careful attention to detail when performing arithmetic operations. Even a small error in arithmetic can have a significant impact on the solution. Therefore, it's essential to double-check calculations and ensure that all operations are performed correctly. This includes paying attention to signs, order of operations, and the properties of numbers. Furthermore, this step reinforces the importance of understanding the relationship between addition and subtraction. Subtraction is the inverse operation of addition, and vice versa. When solving equations, we often use inverse operations to isolate the variable. In this case, subtracting 4 from both sides of the equation is the inverse operation of adding 4. However, if we perform the subtraction incorrectly, we will end up with the wrong equation and the wrong solution.

  • Option 6: 5x = -58

    • This equation is the result of correctly subtracting 4 from both sides of the equation 5x + 4 = -54, as shown in our step-by-step solution. Thus, this option is correct. This step demonstrates the importance of accurately applying inverse operations when solving equations. To isolate the variable x, we need to undo the operations that are being performed on it. In this case, the equation 5x + 4 = -54 involves adding 4 to the term 5x. To undo this addition, we subtract 4 from both sides of the equation. This is a fundamental principle in solving equations: performing the inverse operation on both sides to maintain balance and isolate the variable. Furthermore, this step underscores the significance of understanding the properties of equality. The subtraction property of equality states that if we subtract the same quantity from both sides of an equation, the equality remains true. This property allows us to manipulate equations while preserving their solutions. By applying this property correctly, we can isolate the variable and find its value. Moreover, this step reinforces the concept of equivalent equations. Equivalent equations are equations that have the same solution set. Subtracting 4 from both sides of the equation 5x + 4 = -54 results in an equivalent equation, 5x = -58, which has the same solution for x. This equivalence is crucial because it allows us to transform the equation into a simpler form without changing the solution.

Conclusion

Therefore, the three equations that have the same value of x as the original equation are:

  • 6((5/6)x + (2/3)) = -9(6)
  • 5x + 4 = -54
  • 5x = -58

By systematically manipulating the original equation and comparing it to the options, we identified the equivalent equations. This process underscores the importance of understanding algebraic principles and applying them accurately to solve equations effectively. Remember, the key to solving equations lies in maintaining balance and simplifying expressions while adhering to mathematical properties. With practice and a solid grasp of these concepts, you can confidently tackle a wide range of equation-solving challenges.