Solving The Equation 5(w-7) = 9w-7 A Step-by-Step Guide
Introduction: Decoding the Algebraic Puzzle
In the realm of mathematics, equations serve as intricate puzzles, and solving them is akin to unlocking hidden treasures. Today, we embark on a journey to dissect and conquer the equation 5(w-7) = 9w-7, a seemingly complex expression that holds a straightforward solution. This comprehensive guide aims to demystify the process, providing a step-by-step walkthrough that empowers you to confidently solve similar algebraic challenges. We'll not only focus on arriving at the answer but also delve into the underlying principles, ensuring a deeper understanding of the mathematical concepts involved. Whether you're a student grappling with algebra or simply a curious mind eager to expand your mathematical horizons, this article will equip you with the knowledge and skills to tackle such equations with ease.
Algebraic equations like this one are fundamental building blocks in mathematics and have applications in various fields, from physics and engineering to economics and computer science. Mastering the art of solving equations is therefore a crucial step in any mathematical endeavor. Our approach will be methodical and clear, breaking down each step into manageable components. We'll explore the distributive property, the concept of combining like terms, and the art of isolating the variable. By the end of this journey, you'll not only know the value of 'w' that satisfies the equation but also possess a robust understanding of the techniques used to arrive at that solution. So, let's embark on this mathematical adventure and unlock the secrets hidden within the equation 5(w-7) = 9w-7.
Step 1: Applying the Distributive Property
The first step in solving the equation 5(w-7) = 9w-7 is to simplify the left side by applying the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our case, 'a' is 5, 'b' is 'w', and 'c' is -7. Applying the distributive property, we multiply 5 by both terms inside the parentheses: 5 * w and 5 * -7. This gives us 5w - 35. The equation now becomes:
5w - 35 = 9w - 7
This step is crucial because it eliminates the parentheses, making the equation easier to manipulate. The distributive property is a cornerstone of algebra, and its correct application is essential for solving a wide range of equations. Understanding this principle allows us to break down complex expressions into simpler terms, paving the way for further simplification. Misapplication of the distributive property is a common source of errors in algebra, so it's vital to master this step. Think of it as carefully unwrapping a package – each term inside the parentheses needs to be touched by the multiplier outside. By correctly applying the distributive property, we've transformed our original equation into a more manageable form, setting the stage for the next steps in our solution process. The equation 5w - 35 = 9w - 7 is now a linear equation in one variable, which we can solve using standard algebraic techniques. This initial simplification is a testament to the power of algebraic rules in transforming complex problems into simpler ones.
Step 2: Isolating the Variable 'w'
Our next objective is to isolate the variable 'w' on one side of the equation. This involves strategically moving terms around to achieve the form w = [some value]. Looking at our equation, 5w - 35 = 9w - 7, we observe that 'w' appears on both sides. A common strategy is to gather the 'w' terms on one side and the constant terms on the other. To do this, we can subtract 5w from both sides of the equation. This ensures that we maintain the balance of the equation, as any operation performed on one side must be mirrored on the other. Subtracting 5w from both sides, we get:
5w - 35 - 5w = 9w - 7 - 5w
This simplifies to:
-35 = 4w - 7
Now, we have all the 'w' terms on the right side. Next, we need to isolate the term with 'w' by getting rid of the constant term, which is -7 in this case. To do this, we add 7 to both sides of the equation:
-35 + 7 = 4w - 7 + 7
This simplifies to:
-28 = 4w
We're now very close to isolating 'w'. We have -28 = 4w, which means 4 times 'w' equals -28. To find the value of 'w', we need to undo the multiplication by 4. This is achieved by dividing both sides of the equation by 4. Isolating the variable is a fundamental skill in algebra, and it's crucial to perform these steps with precision, ensuring that each operation maintains the equation's balance.
Step 3: Solving for 'w'
Having successfully isolated the variable 'w' in the previous step, we now stand on the precipice of finding its value. Our equation currently reads -28 = 4w. As we discussed, to solve for 'w', we need to undo the multiplication by 4. This is achieved by dividing both sides of the equation by 4. This operation maintains the balance of the equation while simultaneously isolating 'w'. Performing the division, we get:
-28 / 4 = 4w / 4
This simplifies to:
-7 = w
Therefore, the solution to the equation 5(w-7) = 9w-7 is w = -7. This is the value of 'w' that, when substituted back into the original equation, will make the equation true. Solving for a variable is the ultimate goal in many algebraic problems, and it represents the culmination of our efforts in manipulating and simplifying the equation. The ability to confidently solve for variables is a key skill in mathematics and opens doors to tackling more complex problems in various fields.
To ensure the accuracy of our solution, it's always a good practice to check our answer by substituting it back into the original equation. This verification step provides an added layer of confidence and helps catch any potential errors. In the next section, we'll perform this crucial check to solidify our understanding and ensure the correctness of our solution.
Step 4: Verifying the Solution
In the realm of mathematics, rigor and precision are paramount. Therefore, it's not enough to simply arrive at a solution; we must also verify its correctness. This crucial step involves substituting the value we obtained for 'w' back into the original equation. In our case, we found that w = -7. Let's substitute this value into the original equation, 5(w-7) = 9w-7, and see if it holds true.
Substituting w = -7 into the equation, we get:
5(-7 - 7) = 9(-7) - 7
Now, we simplify both sides of the equation. On the left side, we first perform the operation inside the parentheses:
-7 - 7 = -14
So, the left side becomes:
5(-14)
Multiplying 5 by -14, we get:
-70
Now, let's simplify the right side of the equation. We first multiply 9 by -7:
9(-7) = -63
So, the right side becomes:
-63 - 7
Subtracting 7 from -63, we get:
-70
Comparing both sides, we see that:
-70 = -70
Since both sides of the equation are equal, our solution w = -7 is verified. This confirmation step is essential as it provides us with the assurance that our algebraic manipulations were accurate and that we have indeed found the correct solution. Verification is a cornerstone of mathematical practice, reinforcing the importance of precision and attention to detail.
Conclusion: Mastering the Art of Equation Solving
In this comprehensive guide, we embarked on a journey to solve the equation 5(w-7) = 9w-7. We meticulously dissected the equation, applying the distributive property, isolating the variable, and finally arriving at the solution w = -7. Furthermore, we rigorously verified our solution, ensuring its accuracy and solidifying our understanding of the process. This journey highlights the fundamental principles of algebra and the systematic approach required to solve equations effectively.
The ability to solve equations is a cornerstone of mathematical proficiency. It's a skill that transcends the classroom and finds applications in various real-world scenarios, from financial planning to scientific research. By mastering the techniques outlined in this guide, you've equipped yourself with a valuable tool for problem-solving and critical thinking.
Remember, the key to success in mathematics lies in practice and perseverance. The more you engage with equations and mathematical problems, the more confident and proficient you'll become. Don't be afraid to tackle challenging problems; each attempt, whether successful or not, provides an opportunity to learn and grow. The world of mathematics is vast and fascinating, and with a solid foundation in equation solving, you're well-equipped to explore its many wonders. So, continue to practice, continue to learn, and continue to unravel the mysteries of mathematics.