Solving For V In -6 = 8/(v-7) A Step-by-Step Guide
In the realm of algebra, solving for a variable within an equation is a fundamental skill. This article delves into the step-by-step process of solving for 'v' in the equation -6 = 8/(v-7), providing a comprehensive guide for students and enthusiasts alike. We will explore the underlying principles, demonstrate the algebraic manipulations involved, and arrive at the simplified solution. This exploration will not only solidify your understanding of solving equations but also enhance your problem-solving skills in mathematics.
Understanding the Equation
At the heart of our task lies the equation -6 = 8/(v-7). This equation presents a relationship between the variable 'v' and constants. To solve for v, our primary objective is to isolate 'v' on one side of the equation. This involves a series of algebraic manipulations that maintain the equality of both sides. Understanding the structure of the equation is crucial for devising an effective solution strategy. The equation involves a fraction, which means we'll need to address the denominator (v-7) to isolate 'v'. This typically involves multiplying both sides of the equation by the denominator, a fundamental step in solving equations involving fractions. Furthermore, we'll need to handle the constants (-6 and 8) appropriately to ultimately isolate 'v'. This might involve addition, subtraction, multiplication, or division operations, depending on the specific steps required to simplify the equation.
The process of solving for 'v' is not merely about finding a numerical answer; it's about understanding the underlying mathematical principles that govern equations and their solutions. By carefully analyzing the equation and applying the appropriate algebraic techniques, we can unravel the value of 'v' and gain a deeper appreciation for the beauty and elegance of mathematics. The equation -6 = 8/(v-7) is a classic example of an algebraic equation that requires a systematic approach to solve. It showcases the importance of understanding the order of operations, the properties of equality, and the techniques for manipulating fractions. As we delve into the solution process, we will highlight these key concepts and demonstrate how they come together to produce a solution. The journey of solving this equation is not just about finding 'v'; it's about strengthening your mathematical foundation and building confidence in your ability to tackle more complex problems.
Step-by-Step Solution
1. Eliminate the Fraction
To initiate the solution process, we aim to eliminate the fraction by multiplying both sides of the equation by the denominator (v-7). This crucial step clears the fraction and simplifies the equation, making it easier to manipulate. Multiplying both sides by (v-7) gives us: -6(v-7) = 8. This step is justified by the multiplication property of equality, which states that multiplying both sides of an equation by the same non-zero value preserves the equality. By eliminating the fraction, we transform the equation into a more manageable form, paving the way for isolating the variable 'v'. This process is a common strategy in solving equations involving fractions, and it's essential to master this technique for algebraic problem-solving. The key is to recognize the denominator as a barrier to isolating the variable and to use multiplication as a tool to overcome this barrier. As we move forward, we'll continue to apply algebraic principles to further simplify the equation and ultimately solve for 'v'.
2. Distribute on the Left Side
Next, we distribute the -6 on the left side of the equation, applying the distributive property of multiplication over subtraction. This involves multiplying -6 by both terms inside the parentheses, resulting in -6v + 42 = 8. The distributive property is a fundamental concept in algebra, allowing us to expand expressions and simplify equations. In this case, it enables us to remove the parentheses and create a linear equation that is easier to solve. The key is to multiply the term outside the parentheses by each term inside the parentheses, paying careful attention to the signs. The result, -6v + 42 = 8, represents a significant step towards isolating 'v'. We have now transformed the equation into a form where we can directly apply addition or subtraction to move the constant term to the other side. This step showcases the power of algebraic manipulation in simplifying complex expressions and making them more amenable to solution. By mastering the distributive property, you'll gain a valuable tool for tackling a wide range of algebraic problems.
3. Isolate the Term with 'v'
To isolate the term with 'v', we subtract 42 from both sides of the equation. This step follows the subtraction property of equality, which dictates that subtracting the same value from both sides of an equation maintains the equality. Subtracting 42 from both sides of -6v + 42 = 8 yields -6v = -34. This step effectively moves the constant term to the right side of the equation, leaving the term with 'v' isolated on the left. This is a crucial step in the process of solving for 'v', as it brings us closer to our goal of having 'v' by itself. The subtraction property of equality is a cornerstone of algebraic manipulation, allowing us to rearrange equations while preserving their fundamental balance. By understanding and applying this property, we can systematically isolate variables and solve for their values. The result, -6v = -34, is a simple equation that can be solved in the next step by dividing both sides by the coefficient of 'v'.
4. Solve for 'v'
Finally, to solve for 'v', we divide both sides of the equation by -6. This utilizes the division property of equality, which states that dividing both sides of an equation by the same non-zero value preserves the equality. Dividing both sides of -6v = -34 by -6 gives us v = -34/-6. This step isolates 'v' on the left side, revealing its value. The division property of equality is a fundamental tool in algebra, allowing us to undo multiplication and isolate variables. By carefully applying this property, we can solve for 'v' in a systematic and reliable manner. The result, v = -34/-6, represents the solution to the equation. However, it's important to simplify this fraction to obtain the final answer in its simplest form. This involves finding the greatest common divisor of the numerator and denominator and dividing both by it. In the next step, we'll simplify the fraction to arrive at the ultimate solution for 'v'.
5. Simplify the Answer
The solution, v = -34/-6, can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2. This simplification yields v = 17/3. This is the simplified form of the solution, expressing 'v' as an irreducible fraction. Simplifying fractions is a crucial step in mathematical problem-solving, ensuring that the answer is presented in its most concise and understandable form. In this case, dividing both -34 and -6 by 2 resulted in 17 and 3, respectively, leading to the simplified fraction 17/3. This fraction represents the value of 'v' that satisfies the original equation. It's important to note that simplifying fractions does not change their value; it merely expresses them in a different form. By simplifying our solution, we not only present it in its most elegant form but also make it easier to interpret and use in further calculations. The final answer, v = 17/3, represents the culmination of our step-by-step solution process, showcasing the power of algebraic manipulation in solving equations.
Final Answer
Therefore, the solution to the equation -6 = 8/(v-7) is v = 17/3. This comprehensive guide has demonstrated the step-by-step process of solving for 'v', highlighting the key algebraic principles and techniques involved. From eliminating the fraction to simplifying the final answer, each step has been carefully explained to provide a clear and thorough understanding of the solution. This solution represents the value of 'v' that satisfies the original equation, and it can be verified by substituting it back into the equation and confirming that both sides are equal. The process of solving for 'v' has not only yielded a numerical answer but also reinforced our understanding of algebraic problem-solving strategies. By mastering these techniques, you can confidently tackle a wide range of algebraic equations and further develop your mathematical skills.
Verification (Optional)
To verify our solution, we substitute v = 17/3 back into the original equation: -6 = 8/((17/3)-7). Simplifying the denominator, we get (17/3) - 7 = (17/3) - (21/3) = -4/3. Substituting this back into the equation, we have -6 = 8/(-4/3). Dividing by a fraction is the same as multiplying by its reciprocal, so we get -6 = 8 * (-3/4) = -6. Since both sides of the equation are equal, our solution v = 17/3 is verified. This verification step is crucial in ensuring the accuracy of our solution. By substituting the calculated value of 'v' back into the original equation, we can confirm that it satisfies the equation and that no errors were made during the solution process. This practice not only strengthens our confidence in the solution but also reinforces our understanding of the equation and the relationship between the variable and the constants. The verification step is a valuable tool in algebraic problem-solving, and it's always recommended to perform it whenever possible. In this case, the verification has confirmed that our solution, v = 17/3, is indeed correct.
Solve for v in the equation: -6 = 8/(v-7). Simplify the solution as much as possible.
Solving for v in -6 = 8/(v-7) A Step-by-Step Guide
