Solving For A In The Equation 3a + B = 54 Given B = 9

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of a variable within a linear equation. Specifically, we will tackle the equation 3a + b = 54, where our goal is to determine the value of a given that b = 9. This type of problem is a staple in introductory algebra and serves as a building block for more complex mathematical concepts. By understanding how to solve such equations, you'll be better equipped to tackle a wide range of mathematical challenges. This exploration will not only demonstrate the step-by-step solution but also highlight the underlying principles of algebraic manipulation, ensuring a comprehensive understanding of the topic.

Understanding the Equation

The equation 3a + b = 54 is a linear equation with two variables, a and b. 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 variables in a linear equation can only have a power of one. In our equation, 3a represents three times the variable a, and b represents the second variable. The equation states that the sum of 3a and b equals 54. When we are given a specific value for one of the variables, in this case b = 9, we can substitute this value into the equation and solve for the remaining variable, a. This process of substitution and solving is a core technique in algebra and is used extensively in various mathematical and scientific applications. Understanding the structure of the equation and the relationships between the variables is crucial for finding the correct solution.

Step-by-Step Solution

1. Substitute the value of 'b'

The first step in solving for a is to substitute the given value of b, which is 9, into the equation. This replaces the variable b with its numerical value, simplifying the equation and bringing us closer to isolating a. When we substitute b = 9 into 3a + b = 54, we get 3a + 9 = 54. This substitution is a fundamental technique in algebra, allowing us to reduce the number of unknowns in the equation. By replacing a variable with its known value, we transform the equation into a simpler form that can be more easily solved. This step is crucial for setting up the equation for further algebraic manipulation.

2. Isolate the term with 'a'

Our next goal is to isolate the term containing a, which is 3a. To do this, we need to eliminate the constant term, 9, from the left side of the equation. We achieve this by subtracting 9 from both sides of the equation. This maintains the equality because we are performing the same operation on both sides. Subtracting 9 from both sides of 3a + 9 = 54 gives us 3a + 9 - 9 = 54 - 9, which simplifies to 3a = 45. This step is a crucial application of the properties of equality, ensuring that the equation remains balanced while we isolate the variable term. By isolating 3a, we are one step closer to finding the value of a itself.

3. Solve for 'a'

Now that we have the equation 3a = 45, we can solve for a by dividing both sides of the equation by 3. This isolates a on the left side and gives us its value. Dividing both sides by 3, we get (3a) / 3 = 45 / 3, which simplifies to a = 15. This final step is the culmination of our algebraic manipulations, providing us with the solution to the equation. The value a = 15 satisfies the original equation when b = 9, making it the correct answer. This process of isolating the variable through division is a common technique in solving linear equations and demonstrates the power of algebraic principles.

Verification

To ensure our solution is correct, it's always a good practice to verify it. We can do this by substituting the values of a and b back into the original equation. If the equation holds true, then our solution is correct. Substituting a = 15 and b = 9 into 3a + b = 54, we get 3(15) + 9 = 54. Simplifying this, we have 45 + 9 = 54, which is indeed true. This verification step confirms that our solution of a = 15 is accurate and satisfies the given conditions. Verifying the solution is an important habit in mathematics, as it helps to catch any potential errors and builds confidence in the correctness of the answer.

Conclusion

In conclusion, we have successfully found the value of a in the equation 3a + b = 54 when b = 9. By following a step-by-step approach of substitution, isolation, and solving, we determined that a = 15. This process highlights the fundamental principles of algebraic manipulation and equation-solving. Understanding these principles is essential for tackling more complex mathematical problems. The ability to solve linear equations is a foundational skill that is applicable in various fields, from science and engineering to economics and everyday problem-solving. By mastering these techniques, you can confidently approach a wide range of mathematical challenges and develop a strong foundation in algebra. Remember to always verify your solutions to ensure accuracy and build confidence in your mathematical abilities.