Solving For 'a': A Step-by-Step Guide
In the realm of mathematics, solving for variables is a fundamental skill. This guide will walk you through the process of solving for 'a' in the equation 7a - 2b = 5a + b. We'll break down each step, ensuring clarity and understanding. Let's dive in!
Understanding the Equation
Before we jump into the solution, let's understand the equation we're dealing with: 7a - 2b = 5a + b. This is a linear equation with two variables, 'a' and 'b'. Our goal is to isolate 'a' on one side of the equation to express it in terms of 'b'. This means we want to manipulate the equation until we have 'a = some expression involving b'. The ability to isolate variables like 'a' is crucial in many areas of math and science, allowing us to understand the relationship between different quantities and solve for unknowns in various situations. This foundational skill is not just limited to textbook problems; it's essential for real-world applications, from calculating distances and speeds to modeling complex financial scenarios. By mastering the techniques of solving equations, you're equipping yourself with a powerful tool for problem-solving across a multitude of disciplines. Furthermore, the process of isolating a variable hones your analytical skills, encouraging you to think strategically about how to manipulate equations while adhering to mathematical rules. This kind of critical thinking is invaluable not only in academic pursuits but also in everyday decision-making processes, where you often need to break down complex problems into manageable steps and identify the key factors at play. Therefore, understanding and practicing how to solve for variables is an investment in your overall analytical capabilities, paving the way for success in various fields and aspects of life.
Step 1: Group 'a' terms on one side
The first step is to gather all the terms containing 'a' on one side of the equation. To do this, we can subtract '5a' from both sides of the equation:
7a - 2b - 5a = 5a + b - 5a
This simplifies to:
2a - 2b = b
This step is crucial because it begins the process of isolating 'a', bringing all the 'a' terms together so we can eventually solve for it. By subtracting '5a' from both sides, we maintain the equation's balance, a fundamental principle in algebraic manipulations. Think of an equation like a scale; whatever you do to one side, you must do to the other to keep it balanced. In this case, subtracting '5a' from both sides ensures that the equality remains valid. Moreover, this step demonstrates a core strategy in problem-solving: simplifying complex expressions. By reducing the number of terms on each side of the equation, we make it easier to see the next step and move closer to our goal of isolating 'a'. This strategic approach to simplification is a valuable technique not only in mathematics but also in other areas, where breaking down complex problems into smaller, more manageable parts can lead to clearer solutions. So, this seemingly simple step is actually a demonstration of important mathematical principles and problem-solving strategies that are applicable far beyond the confines of algebra.
Step 2: Isolate the 'a' term
Next, we want to isolate the term with 'a' (which is '2a') on one side. To do this, we need to get rid of the '-2b' term on the left side. We can achieve this by adding '2b' to both sides of the equation:
2a - 2b + 2b = b + 2b
This simplifies to:
2a = 3b
This step is pivotal in isolating 'a' because it eliminates the constant term on the left side, bringing us closer to our final solution. Adding '2b' to both sides is another application of the fundamental principle of maintaining balance in an equation. Just like in Step 1, we're ensuring that the equality remains valid by performing the same operation on both sides. This principle is the bedrock of algebraic manipulation, allowing us to rearrange equations without altering their inherent truth. Furthermore, this step highlights the strategic nature of equation solving. We're not just randomly performing operations; we're deliberately choosing actions that will move us closer to our goal. In this case, adding '2b' was a targeted move to eliminate the '-2b' term and isolate '2a'. This kind of strategic thinking is essential for tackling more complex mathematical problems, where a series of carefully chosen steps is often required to arrive at a solution. Therefore, mastering this step reinforces not only the mechanics of algebra but also the art of problem-solving.
Step 3: Solve for 'a'
Now, we have '2a = 3b'. To solve for 'a', we need to get 'a' by itself. We can do this by dividing both sides of the equation by 2:
(2a) / 2 = (3b) / 2
This simplifies to:
a = (3/2)b
And there we have it! We've successfully solved for 'a'.
This final step is the culmination of our efforts, where we isolate 'a' and express it in terms of 'b', thus achieving the original goal of the problem. Dividing both sides by 2 is, once again, an application of the fundamental principle of maintaining balance in an equation. By performing the same operation on both sides, we ensure that the equality remains valid. This consistent adherence to mathematical rules is crucial for accurate equation solving. Moreover, this step solidifies the understanding of inverse operations. Division is the inverse operation of multiplication, and by dividing both sides by 2, we effectively undo the multiplication of 'a' by 2, leaving 'a' isolated. This concept of inverse operations is fundamental in algebra and other areas of mathematics. Finally, the result, 'a = (3/2)b', provides a clear and concise relationship between 'a' and 'b'. It tells us that the value of 'a' is equal to 3/2 times the value of 'b'. This kind of relationship is invaluable in various mathematical and real-world applications, where understanding the connection between variables is key to solving problems and making predictions. Thus, this final step not only provides the solution but also reinforces important mathematical concepts and their practical implications.
The Solution
Therefore, the solution to the equation 7a - 2b = 5a + b is:
a = (3/2)b
This corresponds to option C in the original problem.
In conclusion, we've successfully navigated the process of solving for 'a' in the given equation, demonstrating the key algebraic principles of grouping like terms, isolating variables, and applying inverse operations. By understanding these concepts and practicing diligently, you can confidently tackle a wide range of mathematical problems. Remember, mathematics is a journey of discovery, and each problem solved is a step forward in your understanding.
For further exploration of algebraic equations and problem-solving techniques, consider visiting resources like Khan Academy Algebra.
