Solve For V Step-by-Step Guide
Linear equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for success in higher-level mathematics. This guide provides a detailed, step-by-step solution to the equation -3(-6v + 8) - 5v = 4(v - 7) - 2, along with explanations and helpful tips to enhance your understanding. In this article, we will meticulously dissect each step involved in solving for the variable v. Linear equations, characterized by their straightforward structure, are the bedrock of numerous mathematical and real-world applications. Proficiency in solving these equations not only bolsters your algebra skills but also equips you to tackle more intricate problems in various domains, including physics, engineering, and economics. Our exploration will emphasize clarity and precision, ensuring that each transformation of the equation adheres to fundamental algebraic principles, thereby guaranteeing the accuracy and validity of our solution. Grasping these principles is paramount, as they form the cornerstone of more advanced mathematical concepts and applications. By the end of this guide, you will not only know the value of v but also possess a deeper insight into the mechanics of equation solving. The ability to manipulate equations with confidence is a valuable asset, opening doors to a broader spectrum of mathematical challenges and analytical opportunities. Let's embark on this journey of mathematical discovery, transforming what may seem like a daunting problem into a clear and conquerable task. By breaking down the equation into manageable steps, we illuminate the path to the solution and reinforce the critical skills necessary for mathematical proficiency.
1. Distribute the Constants
The initial step in solving the equation involves distributing the constants outside the parentheses. This process eliminates the parentheses, making the equation easier to manipulate. We begin by distributing -3 to the terms inside the first set of parentheses (-6v + 8) and then distributing 4 to the terms inside the second set of parentheses (v - 7). This distribution process is a cornerstone of simplifying algebraic expressions and equations, ensuring that each term within the parentheses is correctly accounted for. By meticulously applying the distributive property, we lay the groundwork for subsequent steps, maintaining the integrity of the equation throughout the transformation. The accuracy of this initial distribution is paramount, as any error at this stage can propagate through the rest of the solution. Therefore, we emphasize the importance of careful calculation and double-checking each term to ensure the fidelity of our algebraic manipulation. This meticulous approach not only ensures a correct solution but also instills a valuable habit of precision in mathematical problem-solving. Let's embark on this critical first step, setting the stage for a clear and confident journey toward the solution. By mastering the distributive property, we unlock the door to simplifying complex equations and pave the way for further algebraic exploration.
-3(-6v + 8) - 5v = 4(v - 7) - 2
Distribute -3 to -6v and 8:
(-3 * -6v) + (-3 * 8) - 5v = 4(v - 7) - 2
18v - 24 - 5v = 4(v - 7) - 2
Distribute 4 to v and -7:
18v - 24 - 5v = (4 * v) + (4 * -7) - 2
18v - 24 - 5v = 4v - 28 - 2
2. Combine Like Terms
After distributing the constants, the next crucial step is to combine like terms on each side of the equation. This simplification process consolidates the variable terms and the constant terms, making the equation more manageable and easier to solve. On the left side of the equation, we identify 18v and -5v as like terms, as they both contain the variable v. Similarly, on the right side of the equation, -28 and -2 are like terms because they are both constants. Combining like terms is a fundamental algebraic technique that streamlines equations and reveals the underlying structure, facilitating the isolation of the variable. By meticulously grouping and summing these terms, we reduce the complexity of the equation, paving the way for the subsequent steps in the solution. This process not only simplifies the equation but also enhances clarity, allowing us to focus on the essential elements necessary to solve for v. Let's embark on this vital step, solidifying our understanding of algebraic manipulation and advancing toward the final solution with confidence.
Combine 18v and -5v on the left side:
18v - 5v - 24 = 4v - 28 - 2
13v - 24 = 4v - 28 - 2
Combine -28 and -2 on the right side:
13v - 24 = 4v - 30
3. Isolate the Variable Term
To isolate the variable term, our goal is to gather all terms containing v on one side of the equation. This strategic maneuver simplifies the equation and brings us closer to determining the value of v. In this step, we choose to move the variable term from the right side to the left side, ensuring that all v terms are consolidated in one location. This process involves performing the same operation on both sides of the equation to maintain balance and preserve the equality. By subtracting 4v from both sides, we effectively eliminate the variable term from the right side and transfer it to the left side, where it can be combined with the existing v term. This meticulous manipulation is a cornerstone of algebraic problem-solving, allowing us to systematically isolate the variable and unravel the equation's solution. Let's embark on this critical step, solidifying our understanding of equation manipulation and paving the way for the final determination of v.
Subtract 4v from both sides:
13v - 24 - 4v = 4v - 30 - 4v
9v - 24 = -30
4. Isolate the Constant Term
After isolating the variable term, the next step involves isolating the constant term on the opposite side of the equation. This strategic maneuver further simplifies the equation and brings us closer to determining the value of v. Our objective is to move all constant terms to one side, leaving the variable term isolated on the other side. To achieve this, we perform the inverse operation on both sides of the equation, maintaining balance and preserving the equality. In this case, we add 24 to both sides, effectively eliminating the constant term from the left side and transferring it to the right side, where it can be combined with the existing constant. This meticulous manipulation is a cornerstone of algebraic problem-solving, allowing us to systematically isolate the variable and unravel the equation's solution. Let's embark on this critical step, solidifying our understanding of equation manipulation and paving the way for the final determination of v.
Add 24 to both sides:
9v - 24 + 24 = -30 + 24
9v = -6
5. Solve for V
The final step in solving for v involves isolating the variable by performing the inverse operation on the coefficient of v. This critical maneuver reveals the value of v and completes the solution process. Our objective is to eliminate the coefficient, leaving v by itself on one side of the equation. To achieve this, we divide both sides of the equation by the coefficient of v, which in this case is 9. This division maintains the balance of the equation and isolates v, allowing us to determine its value. This meticulous manipulation is the culmination of our algebraic efforts, providing the final piece of the puzzle and revealing the solution to the equation. Let's embark on this conclusive step, solidifying our understanding of equation manipulation and triumphantly arriving at the value of v.
Divide both sides by 9:
9v / 9 = -6 / 9
v = -6/9
Simplify the fraction:
v = -2/3
Therefore, the solution to the equation is:
v = -2/3
In conclusion, solving the linear equation -3(-6v + 8) - 5v = 4(v - 7) - 2 involves a series of methodical steps, each building upon the previous one. We began by distributing constants to eliminate parentheses, then combined like terms to simplify the equation. Subsequently, we isolated the variable term and the constant term, strategically moving them to opposite sides of the equation. Finally, we solved for v by dividing both sides by its coefficient, revealing the solution v = -2/3. This step-by-step approach underscores the importance of precision and adherence to algebraic principles in solving equations. By mastering these techniques, you gain confidence in your ability to tackle more complex mathematical problems and enhance your overall problem-solving skills. Linear equations are a cornerstone of algebra and mathematics as a whole, and proficiency in solving them opens doors to a wide range of applications in various fields. This journey through the solution process not only provides the answer but also reinforces the critical thinking and analytical skills necessary for success in mathematics and beyond. Embrace the challenge, practice diligently, and you will undoubtedly excel in your mathematical endeavors.
