Solving The Equation 8q - 6 = 2q - 48 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill that unlocks the door to more complex concepts. Equations are mathematical statements asserting the equality of two expressions, and the process of solving them involves finding the value(s) of the variable(s) that make the equation true. This article delves into a step-by-step approach to solving the linear equation 8q - 6 = 2q - 48, providing a clear and concise explanation of the underlying principles and techniques. Whether you're a student grappling with algebra or someone looking to brush up on your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems. Understanding how to manipulate equations and isolate variables is not only crucial for academic success but also has practical applications in various fields, from science and engineering to finance and economics. So, let's embark on this journey of mathematical exploration and unravel the solution to this equation.
Understanding Linear Equations
Before diving into the specifics of our equation, let's establish a solid understanding of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The key characteristic of a linear equation is that the variable(s) appear only to the first power, and there are no products of variables or other more complex functions. Linear equations can involve one or more variables, but in this case, we're dealing with a single-variable linear equation. The general form of a single-variable linear equation is ax + b = c, where 'x' is the variable, and 'a', 'b', and 'c' are constants. Solving a linear equation means finding the value of the variable 'x' that satisfies the equation. This is achieved by performing algebraic operations on both sides of the equation to isolate the variable. These operations include addition, subtraction, multiplication, and division. The goal is to maintain the equality of the equation while simplifying it step by step until the variable is alone on one side. Understanding the properties of equality is essential for solving linear equations effectively. These properties state that performing the same operation on both sides of an equation preserves the equality. With this foundation, we can now approach the equation 8q - 6 = 2q - 48 with confidence.
Step 1: Rearranging the Equation
The initial step in solving the equation 8q - 6 = 2q - 48 involves rearranging the terms to group like terms together. This means bringing all the terms containing the variable 'q' to one side of the equation and the constant terms to the other side. To achieve this, we can subtract 2q from both sides of the equation. This operation maintains the equality because we're performing the same operation on both sides. Subtracting 2q from both sides gives us: 8q - 2q - 6 = 2q - 2q - 48, which simplifies to 6q - 6 = -48. Now, we have all the terms with 'q' on the left side. Next, we need to move the constant term -6 to the right side of the equation. To do this, we add 6 to both sides: 6q - 6 + 6 = -48 + 6. This simplifies to 6q = -42. By performing these two operations, we've successfully rearranged the equation, isolating the term with the variable 'q' on one side and the constant term on the other. This sets the stage for the final step of solving for 'q'. Remember, the key to rearranging equations is to perform the same operations on both sides, ensuring that the equality is preserved throughout the process.
Step 2: Isolating the Variable
After rearranging the equation, we arrive at 6q = -42. The next crucial step is to isolate the variable 'q'. This means getting 'q' by itself on one side of the equation. Currently, 'q' is being multiplied by 6. To undo this multiplication, we need to perform the inverse operation, which is division. We divide both sides of the equation by 6. This maintains the equality of the equation, as we are performing the same operation on both sides. Dividing both sides by 6 gives us: (6q) / 6 = (-42) / 6. On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just 'q'. On the right side, -42 divided by 6 equals -7. Therefore, the equation simplifies to q = -7. We have now successfully isolated the variable 'q' and found its value. This value, q = -7, is the solution to the original equation 8q - 6 = 2q - 48. It is the value that, when substituted back into the original equation, will make the equation true. This step-by-step process of isolating the variable is the cornerstone of solving algebraic equations.
Step 3: Verifying the Solution
Once we've found a solution to an equation, it's always a good practice to verify it. This step ensures that our solution is correct and that we haven't made any errors in our calculations. To verify the solution q = -7 for the equation 8q - 6 = 2q - 48, we substitute -7 for 'q' in the original equation. This gives us: 8(-7) - 6 = 2(-7) - 48. Now, we simplify both sides of the equation separately. On the left side, 8 multiplied by -7 is -56, so we have -56 - 6. This simplifies to -62. On the right side, 2 multiplied by -7 is -14, so we have -14 - 48. This also simplifies to -62. Now, we compare both sides of the equation. We have -62 = -62, which is a true statement. This confirms that our solution, q = -7, is indeed correct. Verifying the solution is a crucial step in the problem-solving process. It not only gives us confidence in our answer but also helps us identify any mistakes we might have made along the way. By substituting the solution back into the original equation, we can ensure that the equation holds true.
Conclusion
In this comprehensive guide, we've meticulously walked through the process of solving the linear equation 8q - 6 = 2q - 48. We began by understanding the fundamentals of linear equations and the properties of equality. We then embarked on a step-by-step solution, starting with rearranging the equation to group like terms, followed by isolating the variable 'q'. Finally, we verified our solution to ensure its accuracy. The solution we found is q = -7. This process highlights the importance of methodical problem-solving in mathematics. Each step, from rearranging terms to isolating the variable, is crucial for arriving at the correct answer. Furthermore, the verification step underscores the value of double-checking our work to avoid errors. Solving equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced topics. The techniques discussed in this article can be applied to a wide range of linear equations, making it a valuable tool for students and anyone working with mathematical concepts. By understanding the underlying principles and practicing regularly, anyone can become proficient in solving equations and tackling mathematical challenges with confidence. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing critical thinking skills.
