Solving For 3x-2 In The Equation (2/3)(9x-6)-4=9x-6
Introduction
In this article, we will delve into the process of solving for the value of 3x-2 given the equation (2/3)(9x-6)-4=9x-6. This is a common type of problem encountered in algebra, and mastering the steps involved is crucial for building a strong foundation in mathematics. Our goal is not just to find the solution but to provide a clear, step-by-step explanation that enhances your understanding of algebraic manipulations and equation-solving techniques. We will break down each step, providing justifications and insights to help you tackle similar problems confidently. This comprehensive guide aims to be a valuable resource for students, educators, and anyone looking to refresh their algebra skills. By the end of this article, you will not only know the value of 3x-2 but also understand the underlying principles that make it possible to arrive at the solution.
Step-by-Step Solution
1. Distribute the fraction
The first step in solving the equation (2/3)(9x-6)-4=9x-6 is to distribute the fraction 2/3 across the terms inside the parentheses. This involves multiplying both 9x and -6 by 2/3. This process is based on the distributive property of multiplication over subtraction, which states that a(b-c) = ab - ac. Applying this property allows us to eliminate the parentheses and simplify the equation, making it easier to work with. Understanding this fundamental principle is key to successfully navigating algebraic manipulations. When we distribute 2/3 to 9x, we get (2/3) * 9x = 6x. Similarly, when we distribute 2/3 to -6, we get (2/3) * -6 = -4. Therefore, the equation becomes 6x - 4 - 4 = 9x - 6. This initial step is crucial as it sets the stage for further simplification and isolation of the variable x. By correctly applying the distributive property, we transform the equation into a more manageable form, paving the way for subsequent steps in the solution process. The result of this step is a simplified equation that is now ready for further manipulation.
2. Combine like terms
After distributing the fraction, our equation looks like this: 6x - 4 - 4 = 9x - 6. The next step involves combining the like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, we have two constant terms, -4 and -4, on the left side of the equation that can be combined. Combining these like terms simplifies the equation and makes it easier to isolate the variable x. This is a fundamental step in solving algebraic equations, as it reduces the complexity of the expression and brings us closer to the solution. By adding -4 and -4, we get -8. Therefore, the equation simplifies to 6x - 8 = 9x - 6. This step is crucial for organizing the equation and making it more manageable. By combining like terms, we have reduced the number of terms in the equation, which will simplify the subsequent steps of isolating the variable and finding its value. The resulting equation, 6x - 8 = 9x - 6, is now in a more streamlined form, ready for the next stage of the solution process.
3. Isolate the variable x
To isolate the variable x, we need to move all terms containing x to one side of the equation and all constant terms to the other side. In the equation 6x - 8 = 9x - 6, we can start by subtracting 6x from both sides of the equation. This will eliminate the x term from the left side and consolidate the x terms on the right side. Subtracting 6x from both sides maintains the equality of the equation, as we are performing the same operation on both sides. This step is crucial for isolating the variable and bringing us closer to the solution. Subtracting 6x from both sides gives us -8 = 3x - 6. Next, we need to isolate the x term further by adding 6 to both sides of the equation. This will eliminate the constant term on the right side and move it to the left side. Adding 6 to both sides maintains the equality of the equation and helps us isolate the variable. Adding 6 to both sides gives us -2 = 3x. Now, we have successfully isolated the x term on one side of the equation. The next step is to solve for x by dividing both sides by the coefficient of x.
4. Solve for x
After isolating the variable term, we have the equation -2 = 3x. To solve for x, we need to divide both sides of the equation by the coefficient of x, which is 3. Dividing both sides by the same number maintains the equality of the equation and allows us to isolate x. This is a fundamental step in solving algebraic equations. Dividing both sides by 3 gives us x = -2/3. This is the value of x that satisfies the original equation. Now that we have found the value of x, we can use it to find the value of the expression 3x - 2. This is the final step in solving the problem. By substituting the value of x into the expression, we can calculate its value and arrive at the final answer. The value x = -2/3 is a critical piece of information that allows us to answer the original question, which is to find the value of 3x - 2.
5. Substitute x into 3x - 2
Now that we have found the value of x to be -2/3, we can substitute this value into the expression 3x - 2 to find its value. This is a straightforward substitution process where we replace the variable x with its numerical value. This step is crucial for answering the original question, which asks for the value of 3x - 2. Substituting x = -2/3 into the expression gives us 3(-2/3) - 2. This expression can be simplified by performing the multiplication first. Multiplying 3 by -2/3 gives us -2. Therefore, the expression becomes -2 - 2. This is a simple subtraction problem that can be easily solved. Subtracting 2 from -2 gives us -4. Therefore, the value of the expression 3x - 2 is -4. This is the final answer to the problem. By substituting the value of x into the expression, we have successfully found its value. This step demonstrates the importance of finding the value of the variable in order to solve for other expressions involving that variable.
Final Answer
After substituting the value of x = -2/3 into the expression 3x - 2, we arrived at the final answer of -4. This means that when x is equal to -2/3, the expression 3x - 2 is equal to -4. This result is a direct consequence of solving the original equation and finding the value of x. The entire process, from distributing the fraction to substituting the value of x, demonstrates the power of algebraic manipulation in solving mathematical problems. The final answer, -4, is the solution to the problem and represents the value of the expression 3x - 2 when x satisfies the given equation. This comprehensive solution provides a clear understanding of the steps involved in solving the problem and highlights the importance of each step in arriving at the final answer. The result is a concrete numerical value that answers the question posed in the problem. Therefore, the value of 3x - 2 is -4.
Conclusion
In conclusion, we have successfully solved for the value of 3x - 2 in the equation (2/3)(9x-6)-4=9x-6. Through a step-by-step process, we distributed the fraction, combined like terms, isolated the variable x, and finally, substituted the value of x into the expression 3x - 2. The final answer we arrived at is -4. This exercise demonstrates the importance of understanding algebraic principles and applying them systematically to solve equations. Each step in the solution process is crucial for arriving at the correct answer. From the initial distribution to the final substitution, every operation must be performed accurately. This comprehensive guide has not only provided the solution but also explained the reasoning behind each step, enhancing your understanding of algebraic problem-solving techniques. By mastering these techniques, you can confidently tackle similar problems in the future. The ability to solve algebraic equations is a valuable skill that is essential for success in mathematics and various other fields. The value of 3x - 2 is -4.
