Solving For A In The Equation 3a + B = 54 When B = 9
Introduction
In the realm of mathematics, equations serve as powerful tools for expressing relationships between variables and constants. Solving equations is a fundamental skill, enabling us to determine the unknown values that satisfy a given relationship. This article delves into the process of solving a specific equation, 3a + b = 54, when b = 9, to unveil the value of 'a'. We will explore the underlying concepts, step-by-step solution, and practical applications of this mathematical exercise. Let's embark on this journey of mathematical discovery!
Understanding the Equation: 3a + b = 54
The equation 3a + b = 54 represents a linear relationship between 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. In this case, 'a' and 'b' are the variables, and 3 and 1 are their respective coefficients. The constant term in the equation is 54. To grasp the essence of this equation, envision it as a balancing act. The left-hand side, 3a + b, must perfectly equal the right-hand side, 54. Our mission is to find the value of 'a' that maintains this balance, given that 'b' is 9. The equation can be interpreted as "three times a number 'a' plus a number 'b' equals 54". When we substitute the value of 'b', we are essentially narrowing down the possibilities for 'a', making the equation easier to solve. The importance of understanding the equation's structure lies in our ability to manipulate it algebraically. We can perform operations on both sides of the equation without disrupting the balance, as long as we apply the same operation to both sides. This principle is crucial for isolating the variable we want to solve for, which in this case is 'a'. The beauty of algebra lies in its ability to transform complex relationships into simpler forms, allowing us to extract the information we need. By understanding the interplay between variables and constants, we can unlock the secrets hidden within equations.
Substituting b = 9 into the Equation
The first step in solving the equation 3a + b = 54 when b = 9 is to substitute the value of 'b' into the equation. Substitution is a fundamental algebraic technique where we replace a variable with its known value. This simplifies the equation, reducing the number of unknowns and bringing us closer to the solution. In this case, we replace 'b' with 9, transforming the equation into: 3a + 9 = 54. By performing this substitution, we have effectively eliminated one variable from the equation, leaving us with a single unknown, 'a'. This is a crucial step because it allows us to isolate 'a' and determine its value. The act of substitution highlights the power of algebraic manipulation. We are not changing the underlying relationship expressed by the equation; rather, we are expressing it in a different form that is more conducive to solving for 'a'. The equation 3a + 9 = 54 now presents a clearer picture. We can see that 3 times 'a', when added to 9, results in 54. This concrete representation of the relationship makes it easier to visualize the steps required to isolate 'a'. Substitution is not just a mechanical process; it's a logical step that simplifies the problem and sets the stage for further algebraic manipulation. It's a testament to the elegance and efficiency of mathematical methods. The ability to substitute values and simplify equations is a cornerstone of algebraic problem-solving, and mastering this technique is essential for success in mathematics and related fields.
Isolating the Term with 'a'
After substituting b = 9 into the equation, we have 3a + 9 = 54. The next crucial step is to isolate the term containing 'a', which is 3a. To achieve this, we need to eliminate the constant term, +9, from the left side of the equation. The principle of algebraic manipulation dictates that we can perform the same operation on both sides of the equation without altering the equality. Therefore, to eliminate +9, we subtract 9 from both sides of the equation: 3a + 9 - 9 = 54 - 9. This operation maintains the balance of the equation while strategically moving us closer to isolating 'a'. Subtracting 9 from both sides simplifies the equation to: 3a = 45. Now, the term containing 'a' is isolated on the left side, and we have a simpler equation to work with. This step demonstrates the power of inverse operations in algebra. Subtraction is the inverse operation of addition, and by applying it strategically, we effectively undo the addition of 9. Isolating the variable term is a fundamental technique in solving equations. It's like peeling away layers to reveal the core element we're interested in. In this case, we're peeling away the constant term to expose the relationship between 3 and 'a'. The equation 3a = 45 now clearly states that 3 times 'a' equals 45. This simplified form makes it much easier to determine the value of 'a' in the subsequent step. The process of isolating the variable term is a cornerstone of algebraic problem-solving. It's a skill that is used repeatedly in various mathematical contexts, and mastering it is essential for success in algebra and beyond.
Solving for 'a'
Having isolated the term with 'a', we now have the equation 3a = 45. To solve for 'a', we need to undo the multiplication by 3. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 3: (3a) / 3 = 45 / 3. This operation maintains the balance of the equation while isolating 'a' on the left side. Dividing both sides by 3 simplifies the equation to: a = 15. Thus, we have successfully solved for 'a' and found its value to be 15. This step showcases the elegance of inverse operations in algebra. Division perfectly reverses the effect of multiplication, allowing us to isolate the variable we're solving for. The solution a = 15 means that if we substitute 15 for 'a' in the original equation, 3a + b = 54, along with b = 9, the equation will hold true. This can be verified by substituting the values: 3(15) + 9 = 45 + 9 = 54. The equation is satisfied, confirming that our solution is correct. Solving for 'a' is the culmination of the algebraic manipulation process. It's the moment where the unknown is revealed, and we have a concrete answer. The value of 'a' is not just a number; it's the solution to a puzzle, the answer to a question posed by the equation. The ability to solve for variables is a fundamental skill in mathematics and has wide-ranging applications in science, engineering, economics, and many other fields. The process of solving for 'a' in this equation exemplifies the power and precision of algebra. It's a testament to the logical steps and techniques that allow us to unravel mathematical relationships and find solutions to complex problems.
Verification of the Solution
To ensure the accuracy of our solution, it is crucial to verify it. Verification involves substituting the calculated value of 'a' back into the original equation, along with the given value of 'b', to check if the equation holds true. In this case, we found that a = 15 when b = 9 in the equation 3a + b = 54. To verify this, we substitute a = 15 and b = 9 into the original equation: 3(15) + 9 = 54. Performing the calculations, we get: 45 + 9 = 54. This simplifies to: 54 = 54. Since the left side of the equation equals the right side, our solution is verified. This process of verification is not just a formality; it's a critical step in problem-solving. It provides a safeguard against errors that may have occurred during the algebraic manipulations. By verifying the solution, we gain confidence in its correctness and ensure that our answer is reliable. Verification is particularly important in complex problems where there are multiple steps involved. A small error in one step can propagate through the rest of the solution, leading to an incorrect final answer. Verification helps us catch these errors and correct them before we move on. The act of verifying a solution also deepens our understanding of the problem. It forces us to revisit the original equation and see how the solution fits within the context of the problem. This can lead to new insights and a more thorough understanding of the mathematical relationships involved. In essence, verification is a quality control measure in problem-solving. It's a way of double-checking our work and ensuring that our solution is accurate and meaningful. By consistently verifying our solutions, we develop a habit of precision and rigor in our mathematical thinking.
Conclusion
In conclusion, we have successfully navigated the process of solving the equation 3a + b = 54 when b = 9. Through a series of logical steps, including substitution, isolating the variable term, and applying inverse operations, we determined that the value of 'a' is 15. We further validated our solution through verification, ensuring its accuracy and reliability. This exercise not only demonstrates the application of fundamental algebraic principles but also highlights the importance of problem-solving strategies such as substitution and the use of inverse operations. The ability to solve equations is a cornerstone of mathematical proficiency, with applications spanning diverse fields, from science and engineering to economics and finance. The skills honed in this endeavor, such as algebraic manipulation, logical reasoning, and attention to detail, are invaluable assets in both academic and professional pursuits. By mastering these techniques, we empower ourselves to tackle complex problems and make informed decisions in a world increasingly driven by data and quantitative analysis. The journey of solving this equation serves as a microcosm of the broader mathematical landscape, where challenges are met with ingenuity, and solutions are unveiled through systematic exploration. As we continue to delve into the realm of mathematics, let us embrace the power of equations and the elegance of their solutions.
