Equations With The Same Solution As (5/6)x + (2/3) = -9
In the realm of algebra, a fundamental task is solving equations. Often, we encounter scenarios where multiple equations might appear different but share the same solution. This article delves into the process of identifying equations that have the same solution as a given equation, specifically $\frac{5}{6}x + \frac{2}{3} = -9$. We will explore various algebraic manipulations and techniques to determine which equations maintain the same value of x. Understanding these concepts is crucial for mastering algebra and its applications in various fields.
Understanding Equivalent Equations
At the heart of this problem lies the concept of equivalent equations. Equivalent equations are equations that, despite their differing appearances, possess the same solution set. This means that the value(s) of the variable(s) that satisfy one equation will also satisfy the other equations. The key to identifying equivalent equations lies in performing algebraic operations that maintain the equality of the equation. These operations include adding or subtracting the same quantity from both sides, multiplying or dividing both sides by the same non-zero quantity, and simplifying expressions. When you're trying to solve for equivalent equations, the process involves identifying operations that preserve the solution set. These operations typically include addition, subtraction, multiplication, and division by a constant on both sides of the equation. The initial equation serves as a baseline, and any manipulation must ensure that the value of x remains consistent across all forms.
For example, consider the equations $x + 2 = 5$ and $2x + 4 = 10$. Both equations have the solution $x = 3$. The second equation is simply the first equation multiplied by 2 on both sides. This illustrates a basic principle: multiplying both sides of an equation by a constant results in an equivalent equation. Similarly, adding or subtracting the same constant from both sides will preserve the solution. However, it's important to note that certain operations, such as squaring both sides or multiplying by an expression that could be zero, may introduce extraneous solutions or eliminate valid solutions, respectively. In these cases, careful consideration is needed to ensure that the resulting equation is truly equivalent to the original.
When faced with the task of determining equivalent equations, it is often beneficial to manipulate the given options to resemble the original equation or to solve each equation individually and compare the solutions. The choice of method depends on the complexity of the equations and personal preference. Regardless of the approach, a solid understanding of algebraic principles is essential for success.
Analyzing the Given Equation: $ rac{5}{6}x + rac{2}{3} = -9$
Our starting point is the equation $\frac{5}{6}x + \frac{2}{3} = -9$. To find equations with the same value of x, we need to manipulate this equation using valid algebraic operations. The first step in dealing with complex equations often involves clearing fractions. This can be achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 6 and 3, and their LCM is 6. Multiplying both sides by 6 will eliminate the fractions and simplify the equation.
This manipulation is a critical step because it transforms the equation into a more manageable form without altering its solution. The new equation will be an equivalent equation, meaning it has the same solution set as the original. This principle is based on the fundamental property of equality, which states that performing the same operation on both sides of an equation preserves the equality. The act of multiplying by the LCM is a strategic application of this property, specifically designed to clear fractions and pave the way for easier manipulation and solution.
Beyond clearing fractions, other common algebraic techniques can be employed to manipulate the equation. These include distributing terms, combining like terms, and isolating the variable. The ultimate goal is to transform the equation into a simpler form that reveals the value of x. However, it is crucial to ensure that each step taken maintains the equivalence of the equation. Any operation that alters the solution set will lead to an incorrect identification of equations with the same value of x. Therefore, a thorough understanding of algebraic principles and careful application of these principles are essential for successfully solving equations and identifying equivalent forms.
Examining the Options
Now, let's analyze the given options to determine which ones have the same solution as our original equation. We'll consider each option individually, applying algebraic principles to see if we can transform them into an equivalent form or if they lead to the same solution for x.
Option 1: $6(\frac{5}{6}x + \frac{2}{3}) = -9$
This option appears to be a manipulation of the original equation. It involves multiplying the left-hand side of the original equation by 6. To maintain equality, we must perform the same operation on both sides. However, the right-hand side remains as -9. This suggests a potential issue with the equivalence of this equation. To verify, we can distribute the 6 on the left-hand side and simplify.
Distributing the 6, we get $6 * (\frac{5}{6}x) + 6 * (\frac{2}{3}) = 5x + 4$. So the equation becomes $5x + 4 = -9$. This equation is different from what we would obtain if we multiplied both sides of the original equation by 6, which would be $5x + 4 = -54$. Therefore, this option does not have the same solution as the original equation.
Option 2: $8(\frac{5}{6}x + \frac{2}{3}) = -9(6)$
This option presents a more complex manipulation. It involves multiplying the left-hand side by 8 and the right-hand side by 6. At first glance, it's not immediately clear whether this equation is equivalent to the original. To determine this, we need to simplify both sides and compare the resulting equation with a valid transformation of the original equation. A systematic approach is necessary to avoid errors.
Let's start by distributing the 8 on the left-hand side: $8 * (\frac{5}{6}x) + 8 * (\frac{2}{3}) = \frac{40}{6}x + \frac{16}{3}$. Simplifying the fractions, we get $\frac{20}{3}x + \frac{16}{3}$. On the right-hand side, we have $-9(6) = -54$. So the equation becomes $\frac{20}{3}x + \frac{16}{3} = -54$. To further analyze this, we can compare it with the equation obtained by multiplying both sides of the original equation by a different constant. This will help us determine if the two equations are indeed equivalent.
Option 3: $5x + 4 = -54$
This option appears to be a simplified form of the original equation after some manipulation. To determine if it's equivalent, we need to perform the same operations on the original equation and see if we arrive at this result. The key is to trace the steps back to the original equation and verify that each step is algebraically sound.
Recall the original equation: $\frac5}{6}x + \frac{2}{3} = -9$. To eliminate the fractions, we can multiply both sides by the least common multiple of 6 and 3, which is 6. This gives us{6}x + \frac{2}{3}) = 6 * (-9)$. Distributing the 6 on the left-hand side, we get $5x + 4 = -54$. This is exactly the same as the given option. Therefore, this option is equivalent to the original equation and has the same solution for x.
Step-by-Step Solution to Find Equivalent Equations
To systematically identify equivalent equations, a step-by-step approach is crucial. This involves manipulating the original equation and comparing the resulting equations with the given options. The process should be meticulous, ensuring that each algebraic operation is valid and preserves the solution set. Here’s a detailed breakdown of the steps:
- Start with the original equation: $\frac{5}{6}x + \frac{2}{3} = -9$
- Clear fractions: Multiply both sides by the least common multiple (LCM) of the denominators, which is 6: $6 * (\frac{5}{6}x + \frac{2}{3}) = 6 * (-9)$
- Distribute: Distribute the 6 on the left-hand side: $5x + 4 = -54$
- Isolate the variable term: Subtract 4 from both sides: $5x = -58$
- Solve for x: Divide both sides by 5: $x = -\frac{58}{5}$
Now that we have the solution for x, we can use this information to check the options. Any equation that simplifies to $x = -\frac{58}{5}$ or can be transformed into the equation $5x + 4 = -54$ is an equivalent equation.
Final Answer
Based on our analysis, the equations that have the same value of x as $\frac{5}{6}x + \frac{2}{3} = -9$ are:
- Option 3: $5x + 4 = -54$
To solidify your understanding, practice solving similar problems and explore different algebraic manipulations. Mastering these techniques will enhance your problem-solving skills and deepen your understanding of algebraic equations. Remember, the key is to maintain the equivalence of the equation by performing valid operations on both sides and carefully analyzing the results.
