Finding The Value Of 'a' In The Equation 5a - 10b = 45

Introduction

In this article, we will delve into the process of finding the value of the variable 'a' in a given algebraic equation. Specifically, we will address the equation 5a - 10b = 45, where the value of 'b' is known to be 3. This type of problem is a fundamental concept in algebra, often encountered in middle school and high school mathematics. Understanding how to solve such equations is crucial for building a strong foundation in algebra and related mathematical disciplines. The process involves substituting the known value of 'b' into the equation, simplifying the expression, and then isolating 'a' to determine its numerical value. This exercise demonstrates the application of basic algebraic principles, including substitution, simplification, and solving for a variable. Furthermore, this concept extends to various real-world applications, making it a valuable skill for problem-solving in diverse contexts.

Step-by-Step Solution

1. Substitute the Value of 'b'

The first step in solving the equation is to substitute the given value of 'b', which is 3, into the equation. This means replacing 'b' with 3 in the equation 5a - 10b = 45. The substitution transforms the equation into 5a - 10(3) = 45. This step is crucial as it eliminates one variable, making the equation solvable for 'a'. By replacing 'b' with its numerical value, we reduce the equation to a single variable problem, which is a standard technique in algebra. The substitution method is widely used in solving systems of equations and other mathematical problems where multiple variables are involved. It allows us to simplify complex expressions and isolate the variable we are interested in finding. Correct substitution is vital for obtaining the accurate solution, as any error in this step will propagate through the rest of the calculation. Therefore, careful attention to detail during substitution is essential for successful problem-solving in algebra and related fields.

2. Simplify the Equation

After substituting 'b' = 3 into the equation, we have 5a - 10(3) = 45. The next step involves simplifying the equation by performing the multiplication. We multiply 10 by 3, which gives us 30. The equation now becomes 5a - 30 = 45. This simplification step is crucial because it reduces the complexity of the equation, making it easier to isolate the variable 'a'. Simplifying expressions by performing arithmetic operations is a fundamental technique in algebra and mathematics in general. It allows us to combine like terms and reduce the equation to its simplest form. The order of operations (PEMDAS/BODMAS) dictates that multiplication should be performed before addition or subtraction, ensuring that the equation is simplified correctly. By simplifying the equation, we prepare it for the next step, which is isolating the variable 'a'. This process of simplification is a cornerstone of algebraic manipulation and is essential for solving equations efficiently and accurately.

3. Isolate the Term with 'a'

To isolate the term containing 'a', which is 5a, we need to eliminate the constant term (-30) from the left side of the equation. We achieve this by adding 30 to both sides of the equation 5a - 30 = 45. This operation maintains the equality of the equation while moving the constant term to the right side. Adding 30 to both sides gives us 5a - 30 + 30 = 45 + 30, which simplifies to 5a = 75. This step is a fundamental application of the properties of equality, which state that performing the same operation on both sides of an equation preserves the equality. Isolating the term with the variable is a critical step in solving equations because it brings us closer to determining the value of the variable. By adding the additive inverse of the constant term, we effectively cancel it out on the left side, leaving only the term with 'a'. This process is widely used in algebra and is essential for solving a wide range of equations. The ability to isolate variables is a key skill in mathematical problem-solving and is crucial for success in algebra and related disciplines.

4. Solve for 'a'

Now that we have the equation 5a = 75, the final step is to solve for 'a'. To do this, we need to isolate 'a' by dividing both sides of the equation by 5, which is the coefficient of 'a'. Dividing both sides by 5 gives us (5a)/5 = 75/5. This operation cancels out the 5 on the left side, leaving us with 'a'. Performing the division on the right side, 75 divided by 5 equals 15. Therefore, the solution to the equation is a = 15. This step demonstrates the use of the multiplicative inverse to isolate the variable. Dividing by the coefficient of the variable is a standard technique in algebra for solving equations. It allows us to undo the multiplication and determine the value of the variable. The solution a = 15 satisfies the original equation, as substituting 15 for 'a' and 3 for 'b' in the equation 5a - 10b = 45 yields a true statement. This final step completes the process of solving for 'a' and provides the numerical value that satisfies the given equation.

Verification

To ensure the accuracy of our solution, it is always a good practice to verify the result. We can do this by substituting the value we found for 'a', which is 15, back into the original equation along with the given value of 'b', which is 3. The original equation is 5a - 10b = 45. Substituting a = 15 and b = 3 gives us 5(15) - 10(3) = 45. Now, we simplify the left side of the equation. 5 multiplied by 15 is 75, and 10 multiplied by 3 is 30. So, the equation becomes 75 - 30 = 45. Subtracting 30 from 75 gives us 45, which means the equation simplifies to 45 = 45. Since both sides of the equation are equal, this confirms that our solution a = 15 is correct. Verification is an essential step in problem-solving because it helps to identify any errors that may have occurred during the solution process. By substituting the solution back into the original equation, we can check if it satisfies the equation and ensure that our answer is accurate. This practice is particularly important in algebra and mathematics in general, where errors can easily arise during complex calculations. Therefore, verification should always be included as part of the problem-solving process to ensure the correctness of the solution.

Conclusion

In conclusion, we have successfully determined the value of 'a' in the equation 5a - 10b = 45 when b = 3. By following a step-by-step approach, we first substituted the value of 'b' into the equation, then simplified the equation, isolated the term with 'a', and finally solved for 'a'. This process involved the application of fundamental algebraic principles, including substitution, simplification, and the properties of equality. The solution we found is a = 15, which we verified by substituting it back into the original equation along with the value of 'b'. This verification step confirmed the accuracy of our solution. Solving algebraic equations like this is a crucial skill in mathematics, with applications in various fields, including science, engineering, and economics. Understanding the steps involved in solving equations and the underlying principles is essential for building a strong foundation in mathematics and for problem-solving in real-world contexts. This exercise demonstrates the importance of careful attention to detail and the systematic application of algebraic techniques to arrive at the correct solution.