Solving -9(a-5) = -18 A Step-by-Step Algebraic Guide

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the step-by-step process of solving a linear equation, providing a comprehensive explanation of each operation involved. We will be dissecting the equation -9(a-5) = -18, illustrating how to isolate the variable 'a' and arrive at the solution. This detailed guide will not only clarify the mechanics of solving this specific equation but also reinforce the underlying principles applicable to a wide range of algebraic problems. Understanding these principles is crucial for building a solid foundation in algebra and progressing to more complex mathematical concepts. From distribution to isolating variables, we'll explore each step in detail, ensuring a clear and thorough understanding of the solution process.

To effectively tackle this equation, we must first understand its structure. The equation -9(a-5) = -18 presents a variable 'a' enclosed within parentheses, multiplied by -9. The goal is to isolate 'a' on one side of the equation to determine its value. This involves a series of algebraic manipulations, each designed to simplify the equation while maintaining its balance. The equal sign acts as a balance point, meaning any operation performed on one side must also be performed on the other to preserve the equality. This principle is paramount in solving equations and ensures that the solution obtained is accurate. Understanding the initial setup and the objective is the first critical step in successfully navigating the solution process. We will break down each step, ensuring clarity and comprehension throughout the process.

Step 1: The Distributive Property

The first step in solving the equation is to apply the distributive property. This property states that a(b + c) = ab + ac. In our equation, -9(a-5) = -18, we need to distribute the -9 across the terms inside the parentheses. This means multiplying -9 by both 'a' and -5. Applying this, we get:

-9 * a = -9a -9 * -5 = +45

Therefore, the equation becomes -9a + 45 = -18. The distributive property is a cornerstone of algebraic manipulation, allowing us to simplify expressions and equations. By correctly applying this property, we eliminate the parentheses, making it easier to proceed with isolating the variable. This step is crucial for transforming the equation into a more manageable form, setting the stage for the subsequent steps in the solution process. Understanding and mastering the distributive property is essential for success in algebra.

Step 2: Isolating the Variable Term

Now that we've distributed the -9, our equation is -9a + 45 = -18. The next step is to isolate the term containing the variable, which in this case is -9a. To do this, we need to eliminate the +45 on the left side of the equation. We can achieve this by subtracting 45 from both sides of the equation. Remember, maintaining balance is key! Subtracting 45 from both sides, we get:

-9a + 45 - 45 = -18 - 45

This simplifies to:

-9a = -63

By subtracting 45 from both sides, we have successfully isolated the variable term -9a. This brings us closer to our goal of isolating 'a' completely. This step highlights the importance of inverse operations in solving equations. Subtraction is the inverse operation of addition, allowing us to effectively cancel out the +45. Isolating the variable term is a crucial milestone in the solution process.

Step 3: Solving for 'a'

We've reached the point where our equation is -9a = -63. The final step is to isolate 'a' completely. Currently, 'a' is being multiplied by -9. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the equation by -9:

-9a / -9 = -63 / -9

This simplifies to:

a = 7

Therefore, the solution to the equation is a = 7. By dividing both sides by -9, we have successfully isolated 'a' and determined its value. This final step demonstrates the power of inverse operations in solving equations. Division is the inverse operation of multiplication, allowing us to effectively cancel out the -9 coefficient. With this, we have completed the solution process, arriving at the value of 'a' that satisfies the original equation.

It's always a good practice to verify your solution to ensure its accuracy. To do this, we substitute the value we found for 'a' (which is 7) back into the original equation: -9(a-5) = -18.

Substituting a = 7, we get:

-9(7-5) = -18

Simplifying the expression inside the parentheses:

-9(2) = -18

Multiplying -9 by 2:

-18 = -18

The equation holds true! This confirms that our solution, a = 7, is correct. Verification is a crucial step in the problem-solving process, providing confidence in the accuracy of the solution. By substituting the value back into the original equation, we can ensure that it satisfies the equality. This process helps to catch any potential errors made during the solution process and reinforces the understanding of the equation's properties. Always take the time to verify your solutions, especially in complex problems.

In conclusion, we have successfully solved the equation -9(a-5) = -18 by following a systematic step-by-step approach. We began by applying the distributive property, then isolated the variable term, and finally solved for 'a'. We also emphasized the importance of verifying the solution to ensure accuracy. This process demonstrates the core principles of solving linear equations, which involve using inverse operations to isolate the variable. Mastering these principles is crucial for success in algebra and beyond. The ability to solve equations is a fundamental skill that is applied in various fields, including science, engineering, and economics. By practicing and understanding the concepts discussed in this article, you can build a strong foundation in equation solving and tackle more complex problems with confidence. Remember, consistent practice and a clear understanding of the underlying principles are the keys to success in mathematics. Continue to explore different types of equations and apply these techniques to further enhance your problem-solving skills. The journey of mathematical discovery is ongoing, and each solved equation is a step forward in your understanding.

By understanding each step and the reasoning behind it, you can confidently tackle similar algebraic problems. Remember, practice makes perfect! The more equations you solve, the more comfortable and proficient you will become. This detailed explanation provides a solid foundation for understanding the principles of solving linear equations. Keep exploring and keep learning!