Solving For U In The Equation -12 = 2u + 2(u + 6)

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In the realm of mathematics, solving for variables is a fundamental skill. This article provides a detailed walkthrough for solving the equation -12 = 2u + 2(u + 6) for the variable u. We will break down the steps, explain the underlying principles, and ensure a clear understanding of how to arrive at the solution. Solving for u involves isolating the variable on one side of the equation. This is achieved by applying algebraic operations to both sides of the equation, maintaining equality throughout the process. The main goal is to simplify the equation step by step until we have u isolated. This process typically involves distributing, combining like terms, and using inverse operations. Understanding the order of operations (PEMDAS/BODMAS) is crucial in solving equations correctly. This ensures we perform operations in the correct sequence, leading to accurate results. Solving for u often involves using the distributive property, which states that a(b + c) = ab + ac. Applying this property correctly helps in simplifying expressions and moving towards isolating the variable. Equations are mathematical statements that assert the equality of two expressions. Solving them involves finding the value(s) of the variable(s) that make the equation true. This skill is essential not only in algebra but also in various fields of science, engineering, and economics. Solving for u is a critical skill in mathematics, and mastering it can open doors to more advanced concepts. Whether you're a student learning algebra or someone looking to refresh your math skills, this comprehensive guide will provide you with a step-by-step approach to solving equations effectively. This article will guide you through the process, ensuring that you not only get the correct answer but also understand the underlying principles.

1. Understanding the Equation: βˆ’12=2u+2(u+6)-12 = 2u + 2(u + 6)

The equation we aim to solve is -12 = 2u + 2(u + 6). The variable u is what we need to isolate and find the value for. The first step in solving for u involves understanding the structure of the equation. Here, we have a linear equation with the variable u appearing on the right side. Our objective is to isolate u on one side of the equation to determine its value. Before diving into the solution, it’s important to recognize the different parts of the equation. We have constants (-12 and 6), coefficients (2), and the variable u. Understanding these components helps in applying the correct algebraic operations. The equation represents a balance, where both sides are equal. Any operation performed on one side must also be performed on the other side to maintain this balance. This principle is fundamental in solving equations. By simplifying the equation step by step, we reduce it to a form where the value of u becomes clear. Each step brings us closer to isolating u, which is our ultimate goal. Recognizing the structure of the equation is the foundational step. From here, we will apply algebraic principles to simplify and eventually solve for u. The ability to dissect and understand the structure of an equation is crucial for successfully solving for the unknown variable. This initial step sets the stage for the subsequent steps in the solution process. A clear understanding of the equation's components and structure ensures a logical and accurate approach to solving for u. The equation is a mathematical statement that two expressions are equal, and our task is to find the value of u that makes this statement true. By understanding the components of the equation, we can better navigate the steps required to isolate and solve for u. This initial understanding is the cornerstone of the entire solution process.

2. Applying the Distributive Property: 2(u+6)2(u + 6)

The next step is to apply the distributive property to the term 2(u + 6). The distributive property states that a(b + c) = ab + ac. Applying the distributive property is a crucial step in simplifying equations that contain parentheses. It allows us to remove the parentheses and combine like terms, making the equation easier to solve. In our case, we need to distribute the 2 across both terms inside the parentheses, which are u and 6. This will help us to eliminate the parentheses and simplify the equation further. By correctly applying the distributive property, we transform the equation into a form that is easier to manipulate and solve. This step is a fundamental algebraic technique that is used extensively in solving various types of equations. The distributive property allows us to handle expressions within parentheses by multiplying the term outside the parentheses with each term inside. This simplifies the equation and brings us closer to isolating the variable u. When applying the distributive property, it is essential to ensure that the multiplication is carried out correctly for each term. This ensures the integrity of the equation and leads to an accurate solution. By distributing the 2 across (u + 6), we eliminate the parentheses and set the stage for combining like terms in the next step. This transformation is a key step in the process of solving for u. The distributive property is a foundational concept in algebra, and mastering it is essential for solving equations efficiently and accurately. This step helps in simplifying complex expressions and makes the equation more manageable. By removing the parentheses, we make it easier to combine like terms and eventually isolate the variable u. Applying the distributive property is a key technique in simplifying algebraic expressions and is a vital step in solving equations. It allows us to remove parentheses and combine like terms, which helps in isolating the variable and finding its value. This step is fundamental to solving a wide range of algebraic problems. The correct application of the distributive property is crucial for simplifying the equation and moving closer to the solution. This ensures that the equation remains balanced and the variable u can be isolated effectively.

Applying the distributive property, we get:

2(u+6)=2βˆ—u+2βˆ—6=2u+122(u + 6) = 2 * u + 2 * 6 = 2u + 12

3. Rewriting the Equation: βˆ’12=2u+2u+12-12 = 2u + 2u + 12

Substituting the result back into the original equation, we get -12 = 2u + 2u + 12. This revised equation is now free of parentheses, making it easier to simplify further. Rewriting the equation after applying the distributive property is a key step in solving for u. This step sets the stage for combining like terms and isolating the variable. The rewritten equation provides a clearer picture of the relationship between the terms and the variable u. This clarity helps in making the subsequent steps more straightforward. By substituting the simplified expression back into the original equation, we maintain the equality and move closer to the solution. This substitution is a critical part of the process. The rewritten equation now has like terms on one side, which we can combine to further simplify the equation. This simplification is essential for isolating u. Rewriting the equation is not just about substituting; it’s about transforming the equation into a more manageable form. This transformation is crucial for the subsequent steps in solving for u. The rewritten equation now allows us to focus on combining the terms involving u and then isolating u itself. This step is a logical progression in the solution process. By rewriting the equation, we create a pathway for simplifying the expression and solving for the unknown variable. This step is a foundational part of algebraic problem-solving. The process of rewriting the equation is a deliberate step that aids in the simplification process. It ensures that we are progressing towards isolating the variable in a methodical manner. Rewriting the equation provides a fresh perspective and makes the next steps in the solution process more apparent. This step is a critical bridge between applying the distributive property and combining like terms.

4. Combining Like Terms: 2u+2u2u + 2u

Now, we combine the like terms on the right side of the equation. Like terms are terms that contain the same variable raised to the same power. In our equation, 2u and 2u are like terms. Combining like terms is a fundamental step in simplifying algebraic expressions. It allows us to reduce the number of terms in the equation, making it easier to solve. In this case, we have two terms involving u on the right side of the equation. By combining them, we can simplify the equation and move closer to isolating u. The process of combining like terms involves adding or subtracting the coefficients of the terms. This step is a direct application of the principles of algebra. Combining like terms helps in reducing the complexity of the equation. This simplification is essential for the subsequent steps in the solution process. By combining the u terms, we consolidate the variable's presence in the equation, making it more manageable. This consolidation is a key step in isolating u. Combining like terms is a standard algebraic technique that is used extensively in solving equations. This step is crucial for simplifying the equation and making it easier to work with. By simplifying the equation, we pave the way for the subsequent steps in the solution process, such as isolating the variable. The ability to recognize and combine like terms is an essential skill in algebra. This skill helps in simplifying expressions and solving equations efficiently. Combining like terms is not just about simplifying; it’s about making the equation more transparent and easier to understand. This transparency helps in solving for u accurately. This step sets the stage for further simplification and eventually isolating the variable u.

2u+2u=4u2u + 2u = 4u

Our equation now becomes:

βˆ’12=4u+12-12 = 4u + 12

5. Isolating the Term with u: Subtracting 12

To isolate the term with u, which is 4u, we need to eliminate the constant term on the right side, which is 12. We do this by subtracting 12 from both sides of the equation. Isolating the term with u is a critical step in solving for u. This involves removing any constants or other terms that are added to or subtracted from the term containing u. By isolating the term with u, we bring the variable closer to being fully isolated and solved for. This step is a fundamental algebraic technique. Subtracting 12 from both sides of the equation maintains the balance of the equation. This ensures that the equality remains true throughout the solution process. This step is a direct application of the properties of equality. Eliminating the constant term on the right side of the equation helps in simplifying the equation and making it easier to solve for u. This simplification is essential for the subsequent steps. The process of isolating the term with u is a methodical approach to solving equations. It involves strategically eliminating terms to bring the variable closer to isolation. Isolating the term with u is not just about subtracting; it’s about strategically manipulating the equation to simplify it. This simplification is a key step in solving for u. By isolating the term with u, we create a pathway for the final step of dividing to solve for u. This step is a logical progression in the solution process. Subtracting the constant term is a deliberate step that aids in the isolation process. It ensures that we are progressing towards solving for the variable in a methodical manner. Isolating the term with u provides a clearer view of the variable and its coefficient. This clarity helps in solving for u accurately.

βˆ’12βˆ’12=4u+12βˆ’12-12 - 12 = 4u + 12 - 12

βˆ’24=4u-24 = 4u

6. Solving for u: Dividing by 4

Finally, to solve for u, we need to divide both sides of the equation by the coefficient of u, which is 4. This will isolate u and give us its value. Solving for u involves isolating the variable completely on one side of the equation. This is the final step in the solution process. By dividing both sides of the equation by the coefficient of u, we achieve this isolation. Dividing both sides of the equation by the coefficient maintains the balance of the equation. This ensures that the equality remains true throughout the solution process. This step is a direct application of the properties of equality. Isolating u is the ultimate goal of solving the equation. This step provides us with the value of the variable that satisfies the equation. The process of solving for u is a systematic approach to finding the value of the variable that makes the equation true. It involves strategically manipulating the equation to isolate u. Solving for u is not just about dividing; it’s about completing the algebraic process to find the value of the variable. This value is the solution to the equation. By isolating u, we provide a clear and concise answer to the problem. This clarity is the culmination of the solution process. Dividing by the coefficient is a deliberate step that aids in the isolation of the variable. It ensures that we have solved for the variable accurately. Solving for u provides a concrete value that can be used in further calculations or applications. This value is the key outcome of the solution process. This step completes the algebraic manipulation and provides the solution to the equation.

rac{-24}{4} = rac{4u}{4}

u=βˆ’6u = -6

7. The Solution

Therefore, the solution to the equation -12 = 2u + 2(u + 6) is u = -6. This solution is the value of u that makes the equation true. The solution represents the value of u that satisfies the original equation. This value is the answer to the problem. The solution u = -6 is the result of a series of algebraic steps. These steps were designed to isolate the variable and find its value. The process of arriving at the solution involved simplifying the equation, combining like terms, and using inverse operations. Each step was crucial in finding the correct value of u. The solution is not just a number; it’s the culmination of a mathematical process. This process is essential for understanding and solving algebraic equations. The solution confirms that our algebraic manipulations were correct and that the equation is balanced. This confirmation is a key aspect of the solution process. The solution provides a concrete answer that can be used in various contexts. This usability is a key characteristic of a successful solution. The solution is the end result of a logical and systematic approach to solving the equation. This approach is essential for solving various algebraic problems. The solution u = -6 is a testament to the power of algebraic techniques. These techniques allow us to solve complex equations and find unknown values. This conclusion marks the end of the solution process and provides the answer to the original problem. It summarizes the entire journey of solving for u. This final answer is the goal of the algebraic manipulation.

Summary

Solving for u in the equation -12 = 2u + 2(u + 6) involves several key steps: applying the distributive property, combining like terms, isolating the variable term, and finally, solving for u. The solution, u = -6, is obtained by methodically applying algebraic principles. Mastering these steps is crucial for solving various algebraic equations effectively. This process highlights the importance of understanding algebraic techniques and applying them systematically to arrive at the correct solution. Each step in the solution process is a building block that contributes to the final answer. This systematic approach is fundamental to solving algebraic problems accurately. The solution is a result of a logical progression of steps. This progression demonstrates the power of algebraic reasoning in solving equations. The ability to solve for variables is a fundamental skill in mathematics. This skill is essential for various applications in science, engineering, and other fields. Solving equations is not just about finding the answer; it’s about understanding the underlying principles of algebra. This understanding is crucial for further mathematical studies. The solution u = -6 is a testament to the effectiveness of algebraic methods. These methods provide a powerful toolkit for solving a wide range of equations. This summary encapsulates the entire solution process and highlights the key concepts involved. It serves as a reminder of the steps and principles used in solving for u. The solution process demonstrates the importance of precision and accuracy in algebraic manipulations. Each step must be performed correctly to arrive at the accurate answer. This final summary reiterates the importance of each step in the solution process. It emphasizes the need for a systematic and logical approach to solving equations.