Solving For A In The Equation 5a - 10b = 45 Given B = 3

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In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of the variable 'a' in a linear equation, given the value of another variable. Specifically, we will address the equation 5a - 10b = 45, where b is known to be 3. This problem is a classic example of algebraic manipulation, a cornerstone of mathematical problem-solving. Understanding how to solve such equations is crucial for various applications, ranging from simple arithmetic problems to more complex scientific and engineering calculations.

Understanding the Equation

The equation 5a - 10b = 45 is a linear equation with two variables, 'a' and 'b'. Linear equations are characterized by their variables being raised to the power of 1, and they form a straight line when graphed. In this case, we are given that b = 3, which simplifies the equation into a single-variable equation, making it solvable for 'a'. The left-hand side of the equation, 5a - 10b, represents a linear combination of 'a' and 'b', while the right-hand side, 45, is a constant. Our goal is to isolate 'a' on one side of the equation to determine its value.

Substituting the Value of b

The first step in solving for 'a' is to substitute the given value of b into the equation. Since we know that b = 3, we can replace 'b' in the equation 5a - 10b = 45 with 3. This substitution gives us 5a - 10(3) = 45. This seemingly simple step is pivotal as it transforms a two-variable equation into a single-variable equation, which is significantly easier to solve. By replacing 'b' with its numerical value, we eliminate one unknown, bringing us closer to finding the value of 'a'. This technique of substitution is widely used in algebra and other branches of mathematics to simplify and solve equations.

Simplifying the Equation

After substituting b = 3 into the equation, we have 5a - 10(3) = 45. The next step is to simplify the equation by performing the multiplication. Multiplying -10 by 3 gives us -30, so the equation becomes 5a - 30 = 45. This simplification is crucial because it combines the constant terms, making it easier to isolate the term containing 'a'. By performing basic arithmetic operations like multiplication, we reduce the complexity of the equation, bringing us closer to the solution. This step is a fundamental aspect of algebraic manipulation, which involves rearranging and simplifying equations to solve for unknowns.

Isolating the Variable 'a'

Now that we have the simplified equation 5a - 30 = 45, our next objective is to isolate the term containing 'a'. This means getting the term 5a by itself on one side of the equation. To do this, we need to eliminate the -30 from the left-hand side. We can accomplish this by adding 30 to both sides of the equation. Adding the same value to both sides maintains the equality, a fundamental principle of algebra. This operation gives us 5a - 30 + 30 = 45 + 30, which simplifies to 5a = 75. Isolating the variable is a crucial step in solving equations, as it brings us closer to determining the value of the unknown.

Solving for 'a'

We have now reached the equation 5a = 75. To finally solve for 'a', we need to isolate 'a' completely. This involves removing the coefficient 5 from the term 5a. Since 5 is multiplying 'a', we can undo this operation by dividing both sides of the equation by 5. Dividing both sides by the same non-zero value maintains the equality. This gives us 5a / 5 = 75 / 5, which simplifies to a = 15. Therefore, the value of 'a' that satisfies the equation 5a - 10b = 45 when b = 3 is 15. This final step demonstrates the power of inverse operations in solving equations, a core concept in algebra.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify the answer. We can do this by substituting the values of 'a' and 'b' back into the original equation. We found that a = 15 and we were given b = 3. Substituting these values into 5a - 10b = 45 gives us 5(15) - 10(3) = 45. Simplifying this, we get 75 - 30 = 45, which further simplifies to 45 = 45. Since the left-hand side equals the right-hand side, our solution is correct. Verification is a crucial step in mathematical problem-solving, as it confirms the validity of the solution and helps prevent errors.

Conclusion

In summary, we have successfully solved for 'a' in the equation 5a - 10b = 45 when b = 3. By following a step-by-step approach, including substitution, simplification, isolation of the variable, and verification, we determined that a = 15. This problem illustrates the fundamental principles of algebraic manipulation, which are essential for solving a wide range of mathematical problems. The ability to solve linear equations is a valuable skill that extends beyond the classroom, finding applications in various fields such as science, engineering, and finance. Mastering these techniques provides a solid foundation for more advanced mathematical concepts and problem-solving.