Solving For 'a' In The Equation 3a + B = 54 Where B Is 9
Introduction
In the realm of mathematics, equations serve as fundamental tools for expressing relationships between variables and constants. Solving these equations allows us to determine the unknown values, providing insights into the underlying mathematical structures. This article delves into the process of solving a simple linear equation, 3a + b = 54, where we aim to find the value of the variable 'a' when the value of 'b' is given as 9. This exploration will not only demonstrate the step-by-step solution but also highlight the core principles of algebraic manipulation and equation-solving techniques. Understanding the concepts of substitution and simplification is crucial in tackling various mathematical problems, and this example serves as a practical illustration of these techniques. The ability to solve linear equations is a foundational skill in mathematics, paving the way for more complex algebraic concepts and applications in diverse fields such as physics, engineering, and economics. This article aims to provide a clear and concise explanation, making the solution accessible to readers with varying levels of mathematical background. By mastering the techniques presented here, readers can confidently approach similar problems and enhance their problem-solving abilities in mathematics. Furthermore, the ability to solve equations is not just limited to academic settings; it is a valuable skill in everyday life, enabling us to make informed decisions and solve practical problems involving numerical relationships. The focus here is not just on finding the answer but also on understanding the process, which is essential for developing a deeper appreciation of mathematical principles and their applications.
Problem Statement: 3a + b = 54, b = 9
The problem we are tackling today involves a linear equation with two variables, 'a' and 'b'. The equation is given as 3a + b = 54. Our objective is to determine the value of 'a', the unknown variable, given that the value of 'b' is known to be 9. This type of problem falls under the category of basic algebra and is a cornerstone of mathematical problem-solving. To solve this equation, we will employ a technique known as substitution, where we replace the variable 'b' with its known value in the equation. This substitution transforms the equation into one with a single variable, 'a', making it easier to isolate and solve for. The process of solving for 'a' involves performing algebraic operations on both sides of the equation to maintain the equality. These operations include subtraction, division, and simplification. Mastering these techniques is crucial for solving a wide range of mathematical problems, including more complex equations and systems of equations. Understanding the underlying principles of algebraic manipulation allows us to approach problems systematically and logically. This particular problem serves as an excellent example of how simple algebraic techniques can be used to solve practical mathematical problems. It also highlights the importance of careful attention to detail and accuracy in mathematical calculations. By working through this problem, we will not only find the value of 'a' but also reinforce our understanding of fundamental algebraic concepts. The ability to solve such equations is essential for success in higher-level mathematics and related fields.
Step-by-Step Solution
To find the value of 'a' in the equation 3a + b = 54, given that b = 9, we will follow a step-by-step approach that emphasizes clarity and understanding. This methodical approach is crucial for solving mathematical problems accurately and efficiently. The first step involves substitution, a fundamental technique in algebra. We replace the variable 'b' in the equation with its given value, which is 9. This substitution transforms the equation from 3a + b = 54 to 3a + 9 = 54. This new equation now involves only one variable, 'a', which simplifies the problem significantly. The next step is to isolate the term containing 'a' on one side of the equation. To do this, we subtract 9 from both sides of the equation. This operation maintains the equality and moves us closer to solving for 'a'. Subtracting 9 from both sides of 3a + 9 = 54 gives us 3a = 54 - 9, which simplifies to 3a = 45. Now that we have isolated the term 3a, the final step is to isolate 'a' itself. To do this, we divide both sides of the equation by 3. This operation effectively cancels out the coefficient of 'a', leaving us with the value of 'a'. Dividing both sides of 3a = 45 by 3 gives us a = 45 / 3, which simplifies to a = 15. Therefore, the value of 'a' in the equation 3a + b = 54, when b = 9, is 15. This step-by-step solution demonstrates the power of algebraic manipulation and the importance of maintaining equality throughout the process. By carefully following these steps, we have successfully solved for the unknown variable 'a'.
Step 1: Substitute the Value of b
As discussed in the previous section, the first critical step in solving for 'a' is to substitute the given value of 'b' into the equation. This substitution is a fundamental technique in algebra that allows us to simplify equations and solve for unknown variables. In our case, we are given the equation 3a + b = 54, and we know that b = 9. Substituting 'b' with 9 in the equation, we get 3a + 9 = 54. This substitution effectively transforms the equation into one with a single variable, 'a', making it easier to solve. The act of substitution is not merely a mechanical replacement; it is a logical step that simplifies the problem by reducing the number of unknowns. It allows us to focus on isolating and solving for the variable of interest, which in this case is 'a'. Understanding the concept of substitution is crucial for tackling a wide range of algebraic problems, including those involving systems of equations and more complex expressions. By mastering this technique, students can confidently approach problems and simplify them into manageable forms. The substitution step also highlights the importance of careful attention to detail. Ensuring that the correct value is substituted for the correct variable is essential for obtaining an accurate solution. A simple mistake in substitution can lead to an incorrect answer, emphasizing the need for precision in mathematical calculations. In summary, the substitution step is a cornerstone of equation-solving, and its correct application is vital for finding the value of unknown variables.
Step 2: Isolate the Term with 'a'
Following the substitution step, the next crucial action in solving for 'a' is to isolate the term containing 'a' on one side of the equation. In our case, the equation after substitution is 3a + 9 = 54. To isolate the term 3a, we need to eliminate the constant term, which is 9, from the left side of the equation. This is achieved by performing the inverse operation on both sides of the equation. Since 9 is added to 3a, the inverse operation is subtraction. We subtract 9 from both sides of the equation to maintain the equality. This gives us 3a + 9 - 9 = 54 - 9, which simplifies to 3a = 45. This step is crucial because it brings us closer to isolating 'a' and determining its value. The principle of maintaining equality by performing the same operation on both sides of the equation is a fundamental concept in algebra. It ensures that the equation remains balanced and that the solution obtained is valid. Mastering this principle is essential for solving various algebraic equations and problems. Isolating the term with the variable of interest is a common strategy in equation-solving. It allows us to focus on the variable and its coefficient, making it easier to solve for the variable. The process of isolating the term also demonstrates the importance of understanding the relationship between addition and subtraction as inverse operations. By applying these operations correctly, we can manipulate equations and solve for unknowns effectively. In this particular step, we have successfully isolated the term 3a, setting the stage for the final step of solving for 'a'.
Step 3: Solve for 'a'
Having successfully isolated the term 3a, the final step in finding the value of 'a' is to solve for 'a' itself. At this stage, we have the equation 3a = 45. To isolate 'a', we need to eliminate the coefficient 3, which is multiplying 'a'. This is achieved by performing the inverse operation, which is division. We divide both sides of the equation by 3 to maintain the equality. This gives us 3a / 3 = 45 / 3. On the left side, the 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with 'a'. On the right side, 45 divided by 3 equals 15. Therefore, the equation simplifies to a = 15. This is the solution to our problem: the value of 'a' in the equation 3a + b = 54, when b = 9, is 15. This step demonstrates the final application of algebraic manipulation to solve for the unknown variable. Understanding the concept of inverse operations is crucial for this step, as we use division to undo the multiplication by 3. The process of solving for 'a' also highlights the importance of accuracy in calculations. Ensuring that the division is performed correctly is essential for obtaining the correct solution. The solution a = 15 is not just a numerical answer; it is a value that satisfies the original equation when substituted back into it. We can verify this by plugging a = 15 and b = 9 into the original equation: 3(15) + 9 = 45 + 9 = 54, which confirms that our solution is correct. In conclusion, by carefully following the steps of substitution, isolation, and solving, we have successfully determined the value of 'a' in the given equation.
Conclusion
In conclusion, we have successfully determined the value of 'a' in the equation 3a + b = 54, given that b = 9. Through a step-by-step approach, we first substituted the value of 'b' into the equation, then isolated the term containing 'a', and finally solved for 'a'. The result of this process is a = 15. This exercise has demonstrated the fundamental principles of algebraic manipulation, including substitution, isolation, and the use of inverse operations. These techniques are essential tools in mathematics and are applicable to a wide range of problem-solving scenarios. Mastering these concepts not only enhances our ability to solve equations but also deepens our understanding of mathematical relationships and structures. The process of solving this equation has also highlighted the importance of accuracy and attention to detail in mathematical calculations. A simple mistake in any step can lead to an incorrect solution, emphasizing the need for careful and methodical work. Furthermore, the ability to solve linear equations is a foundational skill that paves the way for more advanced mathematical concepts and applications. It is a skill that is not only valuable in academic settings but also in everyday life, where we often encounter situations that require problem-solving and analytical thinking. By understanding the principles and techniques demonstrated in this article, readers can confidently approach similar problems and develop their mathematical abilities. The solution presented here is not just an answer; it is a demonstration of the power of algebraic thinking and the importance of a systematic approach to problem-solving.
FAQ
What is the value of a if b is 9?
If b = 9, then the value of a in the equation 3a + b = 54 is 15. We arrive at this solution by substituting 9 for b in the equation, which gives us 3a + 9 = 54. Subtracting 9 from both sides gives 3a = 45, and dividing both sides by 3 gives a = 15. This demonstrates the application of algebraic principles to solve for an unknown variable.
Can this method be applied to other equations?
Yes, the method of substitution and algebraic manipulation can be applied to a wide range of equations, especially linear equations. The key is to identify the known values, substitute them into the equation, and then use inverse operations to isolate the variable you want to solve for. Understanding these techniques provides a versatile approach to problem-solving in mathematics.
Why is it important to isolate the variable?
Isolating the variable is a crucial step in solving equations because it allows us to determine the value of that variable. By isolating the variable, we effectively separate it from other terms and constants, making it possible to find its numerical value. This process involves using inverse operations to undo any operations that are being performed on the variable, ultimately leading to a solution.
