Solving For X In The Equation 3(x+2) = 8(x+3) + 2
In this article, we will delve into the process of solving for x in the equation 3(x+2) = 8(x+3) + 2. This is a fundamental algebraic problem that requires a step-by-step approach to isolate the variable x and determine its value. Understanding how to solve such equations is crucial for various mathematical and scientific applications. This exploration will not only provide the solution to this specific equation but also reinforce the general principles of algebraic manipulation. We will break down each step, ensuring a clear understanding of the logic and techniques involved. Mastering these techniques is essential for anyone pursuing further studies in mathematics, physics, engineering, or any field that relies on mathematical problem-solving. The ability to confidently tackle algebraic equations forms the bedrock of more advanced mathematical concepts. Furthermore, the methodical approach used here can be applied to a wide range of similar problems, making this a valuable skill for both academic and practical endeavors. Let's embark on this journey of algebraic exploration and unravel the solution to this intriguing equation.
Step-by-Step Solution
1. Distribute the constants
The first step in solving the equation 3(x+2) = 8(x+3) + 2 involves distributing the constants on both sides of the equation. This means multiplying the numbers outside the parentheses by each term inside the parentheses. On the left side, we multiply 3 by both x and 2, resulting in 3x + 6. On the right side, we multiply 8 by both x and 3, resulting in 8x + 24. The equation now becomes: 3x + 6 = 8x + 24 + 2. This distribution step is crucial as it eliminates the parentheses, allowing us to combine like terms and proceed with isolating x. A clear understanding of the distributive property is fundamental in algebra, and this step exemplifies its practical application. By correctly distributing the constants, we pave the way for simplifying the equation and ultimately finding the value of x. This initial transformation is a cornerstone of algebraic manipulation, and its accurate execution is essential for the subsequent steps. The distributed form of the equation provides a clearer picture of the relationship between the terms and sets the stage for further simplification.
2. Simplify the equation
After distributing the constants, the equation 3x + 6 = 8x + 24 + 2 can be simplified by combining like terms. On the right side of the equation, we have two constant terms: 24 and 2. Adding these together gives us 26. Thus, the equation simplifies to 3x + 6 = 8x + 26. This simplification step is essential for making the equation more manageable and easier to solve. By combining the constants, we reduce the number of terms in the equation, which streamlines the process of isolating x. Simplifying equations is a core skill in algebra, allowing us to work with more concise expressions. This step not only makes the equation less cumbersome but also brings us closer to the ultimate goal of finding the value of x. The simplified form of the equation allows for a clearer view of the relationship between the variable terms and the constant terms, which is crucial for the next steps in the solution process. This step highlights the importance of identifying and combining like terms in algebraic equations.
3. Isolate the variable terms
To isolate the variable terms in the equation 3x + 6 = 8x + 26, we need to move all the x terms to one side of the equation. A common strategy is to subtract the smaller x term from both sides to keep the coefficient of x positive. In this case, we subtract 3x from both sides of the equation. This gives us: 3x + 6 - 3x = 8x + 26 - 3x. Simplifying this, we get 6 = 5x + 26. By subtracting 3x from both sides, we effectively eliminate the x term from the left side of the equation and consolidate the x terms on the right side. This step is a crucial maneuver in solving algebraic equations, as it brings us closer to isolating the variable x. The principle of maintaining equality by performing the same operation on both sides is fundamental to algebraic manipulation. This step demonstrates the practical application of this principle, ensuring that the equation remains balanced while we strategically move terms around. Isolating the variable terms is a key milestone in the solution process, paving the way for isolating the variable itself.
4. Isolate the constant terms
Now that we have the equation 6 = 5x + 26, the next step is to isolate the constant terms on one side of the equation. To do this, we subtract 26 from both sides of the equation. This gives us: 6 - 26 = 5x + 26 - 26. Simplifying this, we get -20 = 5x. By subtracting 26 from both sides, we effectively eliminate the constant term from the right side of the equation, leaving us with just the term containing x. This step is crucial for isolating x and ultimately solving for its value. The principle of maintaining equality by performing the same operation on both sides is once again applied here, ensuring that the equation remains balanced throughout the process. Isolating the constant terms is a critical step in solving algebraic equations, as it brings us closer to the final solution. This step highlights the importance of performing inverse operations to isolate the variable of interest. By strategically subtracting the constant term, we set the stage for the final step of dividing to solve for x.
5. Solve for x
Finally, to solve for x in the equation -20 = 5x, we need to isolate x completely. To do this, we divide both sides of the equation by the coefficient of x, which is 5. This gives us: -20 / 5 = 5x / 5. Simplifying this, we get x = -4. Therefore, the solution to the equation 3(x+2) = 8(x+3) + 2 is x = -4. This final step is the culmination of all the previous steps, where we systematically manipulated the equation to isolate x. The operation of dividing both sides by the coefficient of x is the inverse operation of multiplication, allowing us to undo the multiplication and reveal the value of x. This step underscores the importance of understanding inverse operations in solving algebraic equations. By correctly performing this division, we arrive at the final solution, which is the value of x that satisfies the original equation. This process demonstrates the power of algebraic manipulation in solving for unknowns and highlights the logical progression of steps required to arrive at a solution.
Conclusion
In conclusion, by following a step-by-step approach, we have successfully solved the equation 3(x+2) = 8(x+3) + 2 and found that x = -4. This process involved distributing constants, simplifying the equation, isolating variable terms, isolating constant terms, and finally, solving for x. Each step is crucial and relies on fundamental algebraic principles. Mastering these principles is essential for solving more complex equations and tackling mathematical problems in various fields. The ability to confidently manipulate algebraic expressions is a valuable skill that extends beyond the classroom and into real-world applications. This exercise not only provides the solution to a specific problem but also reinforces the general methodology for solving algebraic equations. The step-by-step approach demonstrated here can be applied to a wide range of similar problems, making this a valuable learning experience. The journey from the initial equation to the final solution highlights the power of systematic problem-solving and the elegance of algebraic techniques. The solution x = -4 is not just a number; it is the result of a logical progression of steps, each building upon the previous one to unravel the unknown. This underscores the importance of understanding the underlying principles of algebra and the ability to apply them effectively.
