Solving (2/7)a + 20 = 22 A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of solving the algebraic equation $\frac{2}{7}a + 20 = 22$. This equation involves a variable, a, which we aim to isolate and determine its value. Understanding how to solve such equations is a fundamental skill in algebra and is crucial for various mathematical and real-world applications. This guide will break down the solution process into manageable steps, ensuring clarity and a thorough understanding of the underlying principles. By mastering this technique, you'll enhance your problem-solving abilities and gain confidence in tackling more complex algebraic challenges. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide you with the necessary knowledge and techniques to confidently solve equations of this nature. So, let's embark on this algebraic journey and unlock the value of 'a'!

Understanding the Equation

At its core, the equation $\frac{2}{7}a + 20 = 22$ represents a balance. The left side of the equation, $\frac{2}{7}a + 20$, is equivalent to the right side, which is 22. Our goal is to find the value of the variable 'a' that maintains this balance. To achieve this, we need to isolate 'a' on one side of the equation. This involves performing a series of algebraic operations that maintain the equality. Think of it as a balancing scale: whatever operation you perform on one side, you must also perform on the other to keep the scale balanced. The equation consists of several key components: a variable ('a'), a coefficient ($\frac{2}{7}$), a constant (20), and the result (22). Understanding these components is crucial for effectively solving the equation. The coefficient $\frac{2}{7}$ is multiplied by the variable 'a', and the constant 20 is added to this product. The result of this operation is equal to 22. By carefully manipulating these components, we can systematically isolate 'a' and determine its value. This process not only provides the solution but also enhances your understanding of algebraic principles and their practical applications.

Step 1: Isolating the Term with the Variable

The first crucial step in solving the equation $\frac{2}{7}a + 20 = 22$ is to isolate the term that contains the variable 'a'. In this case, the term we need to isolate is $\frac{2}{7}a$. To do this, we need to eliminate the constant term, which is 20, from the left side of the equation. The operation that undoes addition is subtraction. Therefore, we will subtract 20 from both sides of the equation. This maintains the balance of the equation, as we are performing the same operation on both sides. Subtracting 20 from the left side effectively cancels out the +20, leaving us with just $\frac{2}{7}a$. On the right side, we subtract 20 from 22, which gives us 2. So, after this step, our equation transforms from $\frac{2}{7}a + 20 = 22$ to $\frac{2}{7}a = 2$. This seemingly simple step is a critical foundation for the rest of the solution process. By isolating the term with the variable, we have simplified the equation and brought ourselves one step closer to finding the value of 'a'. This process highlights the importance of inverse operations in algebra, which are essential for manipulating equations and solving for unknown variables.

Step 2: Solving for the Variable

Now that we have successfully isolated the term with the variable, our equation stands as $\frac{2}{7}a = 2$. The next step is to isolate the variable 'a' itself. Currently, 'a' is being multiplied by the fraction $\frac{2}{7}$. To undo this multiplication, we need to perform the inverse operation, which is division. However, dividing by a fraction can be a bit tricky. Instead, we can multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of $\frac{2}{7}$ is $\frac{7}{2}$. When we multiply a fraction by its reciprocal, the result is 1, effectively canceling out the fraction and leaving us with just 'a'. So, we multiply both sides of the equation by $\frac{7}{2}$. On the left side, $\frac{7}{2} * \frac{2}{7}a$ simplifies to 1 * a, which is simply 'a'. On the right side, we have 2 * $\frac{7}{2}$. To multiply a whole number by a fraction, we can treat the whole number as a fraction with a denominator of 1. So, we have $\frac{2}{1} * \frac{7}{2}$. Multiplying the numerators (2 * 7) gives us 14, and multiplying the denominators (1 * 2) gives us 2. This results in the fraction $\frac{14}{2}$, which simplifies to 7. Therefore, after this step, we find that a = 7. This is the solution to our equation, meaning that when 'a' is equal to 7, the equation $\frac{2}{7}a + 20 = 22$ holds true. This process demonstrates the power of using inverse operations and reciprocals to solve algebraic equations.

Step 3: Verification

After finding a potential solution to an equation, it's always a good practice to verify the solution. This step ensures that our calculations are correct and that the value we found for the variable actually satisfies the original equation. In our case, we found that a = 7 is the solution to the equation $\frac{2}{7}a + 20 = 22$. To verify this, we will substitute 7 for 'a' in the original equation and check if both sides of the equation are equal. Substituting a = 7 into the equation, we get $\frac{2}{7} * 7 + 20$. First, we perform the multiplication: $\frac{2}{7} * 7$. We can treat 7 as a fraction with a denominator of 1, so we have $\frac{2}{7} * \frac{7}{1}$. Multiplying the numerators (2 * 7) gives us 14, and multiplying the denominators (7 * 1) gives us 7. This results in the fraction $\frac{14}{7}$, which simplifies to 2. So, $\frac{2}{7} * 7$ equals 2. Now, we substitute this back into our equation: 2 + 20. Adding these two numbers gives us 22. Therefore, the left side of the equation, $\frac{2}{7} * 7 + 20$, equals 22, which is the same as the right side of the original equation. This confirms that our solution, a = 7, is correct. Verification is a crucial step in problem-solving, as it provides confidence in the accuracy of the solution and helps prevent errors. By verifying our solution, we can be sure that we have correctly solved the equation and found the true value of the variable.

Conclusion

In conclusion, we have successfully solved the equation $\frac{2}{7}a + 20 = 22$ by following a step-by-step approach. First, we isolated the term with the variable by subtracting 20 from both sides of the equation. This gave us $\frac{2}{7}a = 2$. Next, we solved for the variable 'a' by multiplying both sides of the equation by the reciprocal of $\frac{2}{7}$, which is $\frac{7}{2}$. This yielded the solution a = 7. Finally, we verified our solution by substituting a = 7 back into the original equation, confirming that both sides of the equation were equal. This comprehensive process demonstrates the key principles of solving algebraic equations, including the use of inverse operations and reciprocals. By mastering these techniques, you can confidently tackle a wide range of algebraic problems. Remember, the key to success in algebra is practice and a clear understanding of the underlying concepts. With consistent effort and a methodical approach, you can develop your problem-solving skills and excel in mathematics. This equation, while seemingly simple, provides a solid foundation for more complex algebraic challenges. So, keep practicing, keep exploring, and keep unlocking the power of algebra!