Solving For U A Comprehensive Guide To Algebraic Equations

In the realm of mathematics, algebraic equations form the bedrock of problem-solving. They are the tools we use to decipher the relationships between quantities and to find unknown values. At its core, algebra is a language, and like any language, it has its rules and syntax. To become fluent, we need to understand these rules and practice applying them. This article delves into the process of solving for u in the equation 3u - 9 + 3(3u + 2) = -3(u + 3), offering a step-by-step guide to simplify the equation and isolate the variable. This process involves the application of several fundamental algebraic principles, including the distributive property, combining like terms, and inverse operations. Understanding these concepts is crucial not only for solving this particular equation but also for tackling a wide array of algebraic problems. The ability to solve algebraic equations is a fundamental skill with applications extending far beyond the classroom, from everyday budgeting and financial planning to complex scientific and engineering calculations. By mastering these skills, we empower ourselves to approach problems logically and analytically, fostering critical thinking and problem-solving abilities that are valuable in all aspects of life.

To effectively tackle this equation, we first need to break it down into its constituent parts. The equation 3u - 9 + 3(3u + 2) = -3(u + 3) presents a mix of terms involving the variable u, constant terms, and expressions within parentheses. The presence of parentheses indicates that we'll need to apply the distributive property, a key concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. This step is essential for simplifying the equation and preparing it for further manipulation. Each term in the equation plays a crucial role, and understanding their relationship is paramount to finding the correct solution. The left side of the equation contains terms that need to be simplified before we can isolate u, while the right side offers another set of terms that will interact with the left side during the solving process. Recognizing the structure of the equation, identifying like terms, and understanding the order of operations are all crucial skills that we will employ as we move forward. This initial analysis sets the stage for a systematic approach to solving the equation, ensuring that each step is logical and purposeful.

Now, let's embark on the journey of solving the equation step by step. Our goal is to isolate u on one side of the equation, revealing its value. Each step we take is guided by the fundamental principles of algebra, ensuring that we maintain the balance of the equation. Let's start:

1. Apply the Distributive Property:

The distributive property is our first tool in simplifying the equation. It states that a(b + c) = ab + ac. Applying this to our equation, we get:

3u - 9 + 3(3u + 2) = -3(u + 3)

3u - 9 + 9u + 6 = -3u - 9

Here, we multiplied 3 by both 3u and 2, and -3 by both u and 3, effectively removing the parentheses.

2. Combine Like Terms:

Next, we identify and combine like terms on each side of the equation. Like terms are those that contain the same variable raised to the same power (e.g., 3u and 9u) or are constants (e.g., -9 and 6). Combining them simplifies the equation further:

(3u + 9u) + (-9 + 6) = -3u - 9

12u - 3 = -3u - 9

We've now consolidated the terms on the left side, making the equation more manageable.

3. Isolate the Variable Term:

Our next goal is to gather all the terms containing u on one side of the equation. To do this, we'll add 3u to both sides:

12u - 3 + 3u = -3u - 9 + 3u

15u - 3 = -9

This step moves the u term from the right side to the left, bringing us closer to isolating u.

4. Isolate the Constant Term:

Now, we want to isolate the u term completely by moving the constant term to the other side. We'll add 3 to both sides:

15u - 3 + 3 = -9 + 3

15u = -6

We've now isolated the term with u on the left side.

5. Solve for u:

Finally, to solve for u, we need to undo the multiplication by 15. We do this by dividing both sides by 15:

15u / 15 = -6 / 15

u = -6/15

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

u = -2/5

Thus, we've found the value of u that satisfies the equation.

Before declaring victory, it's crucial to verify our solution. This step involves substituting the value we found for u back into the original equation. If both sides of the equation are equal after the substitution, our solution is correct. This process not only confirms our answer but also reinforces our understanding of the equation and the steps we took to solve it. It's a valuable practice that helps prevent errors and builds confidence in our problem-solving abilities. Let's substitute u = -2/5 into the original equation:

3u - 9 + 3(3u + 2) = -3(u + 3)

3(-2/5) - 9 + 3(3(-2/5) + 2) = -3(-2/5 + 3)

Now, we simplify each side of the equation following the order of operations. This involves performing the multiplication and addition within the parentheses first, then completing the remaining operations.

-6/5 - 9 + 3(-6/5 + 2) = -3(-2/5 + 15/5)

-6/5 - 45/5 + 3(-6/5 + 10/5) = -3(13/5)

-51/5 + 3(4/5) = -39/5

-51/5 + 12/5 = -39/5

-39/5 = -39/5

Since both sides of the equation are equal, our solution u = -2/5 is indeed correct. This verification step provides assurance that we have accurately solved the equation and reinforces our understanding of the algebraic principles involved.

In conclusion, solving for u in the equation 3u - 9 + 3(3u + 2) = -3(u + 3) has been a journey through the fundamental principles of algebra. We've applied the distributive property, combined like terms, and used inverse operations to isolate the variable. Along the way, we've emphasized the importance of each step, highlighting how each operation contributes to the overall goal of finding the solution. The ability to solve algebraic equations is a powerful skill that extends far beyond the classroom. It's a tool that enables us to analyze problems logically, identify relationships between quantities, and find solutions in a systematic way. Whether it's balancing a budget, calculating the trajectory of a rocket, or designing a new structure, the principles of algebra are at play. By mastering these skills, we empower ourselves to tackle a wide range of challenges and to make informed decisions in all aspects of our lives. The solution u = -2/5 is not just a number; it's a testament to the power of algebraic problem-solving. It represents the culmination of a logical process, a journey from a complex equation to a simple answer. As we continue our mathematical journey, the skills we've honed in solving this equation will serve as a foundation for tackling even more complex problems. The key is to practice, to persevere, and to embrace the challenges that mathematics presents.