Solving For X In The Equation 3(x-3) + 2x = -29

In the realm of mathematics, solving for variables is a fundamental skill. One common type of problem involves linear equations, where the goal is to isolate the variable on one side of the equation. In this article, we will delve into the step-by-step process of solving for x in the equation 3(x-3) + 2x = -29. This problem is a classic example of a linear equation, requiring the application of algebraic principles such as the distributive property, combining like terms, and inverse operations. By understanding these techniques, you'll be well-equipped to tackle a wide range of algebraic problems. The ability to solve for variables is not only crucial in mathematics but also in various fields such as physics, engineering, and economics, where equations are used to model real-world phenomena. Mastering these skills will undoubtedly enhance your problem-solving capabilities and provide a solid foundation for more advanced mathematical concepts. We will break down each step in detail, ensuring clarity and comprehension for learners of all levels. From understanding the initial equation to arriving at the final solution, this guide aims to provide a comprehensive understanding of the process involved in solving linear equations. By following along with the explanations and examples, you will gain the confidence and skills needed to solve similar problems independently. This article serves as a valuable resource for students, educators, and anyone looking to strengthen their algebraic abilities. Let's embark on this mathematical journey and unlock the secrets to solving for x.

The first step in solving the equation 3(x-3) + 2x = -29 is to apply the distributive property. The distributive property states that a(b + c) = ab + ac. In our equation, we need to distribute the 3 across the terms inside the parentheses (x-3). This means we multiply 3 by x and 3 by -3. The distributive property is a cornerstone of algebra, enabling us to simplify expressions and equations by removing parentheses. Understanding and applying this property correctly is crucial for solving equations and performing algebraic manipulations. It's a fundamental concept that forms the basis for more advanced algebraic techniques. When dealing with equations involving parentheses, the distributive property is often the first tool we reach for. It allows us to break down complex expressions into simpler, more manageable forms. In this particular equation, distributing the 3 across (x-3) will pave the way for combining like terms and isolating the variable x. This initial step sets the stage for the rest of the solution process. By mastering the distributive property, you'll be well-prepared to tackle a wide range of algebraic problems. Let's proceed with the distribution and see how it simplifies our equation. Remember, the key is to multiply the term outside the parentheses by each term inside the parentheses, ensuring that we maintain the correct signs and coefficients. This step is not only important for solving this specific equation but also for developing a solid understanding of algebraic manipulation in general. With this foundation in place, we can move on to the next step with confidence.

Applying the distributive property, we get:

3(x-3) = 3 * x + 3 * (-3) = 3x - 9

So, the equation becomes:

3x - 9 + 2x = -29

After applying the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 3x - 9 + 2x = -29, we have two terms with x: 3x and 2x. We can combine these terms by adding their coefficients. Combining like terms is a fundamental technique in algebra that simplifies equations and makes them easier to solve. It's like sorting and grouping similar objects together to make them more manageable. In this context, we're grouping the terms with the variable x to simplify the left side of the equation. This step is crucial for isolating the variable and ultimately solving for its value. Without combining like terms, the equation would remain cluttered and more difficult to work with. This process not only simplifies the equation but also brings us closer to the goal of isolating x. It's a necessary step in the overall strategy of solving linear equations. By combining like terms, we're essentially condensing the equation into a more compact and solvable form. This simplification is a key part of algebraic manipulation and is used extensively in various mathematical contexts. As we move forward, we'll see how this step helps us to isolate x and find its value. Remember, the goal is to gather all the terms with x together, which makes it easier to perform the necessary operations to solve for x. Let's combine the like terms and see how our equation transforms.

Combining 3x and 2x, we get:

3x + 2x = 5x

So, the equation now becomes:

5x - 9 = -29

Now that we've combined like terms, our equation is 5x - 9 = -29. The next step is to isolate the variable term, which in this case is 5x. To do this, we need to get rid of the constant term, -9, on the left side of the equation. We can achieve this by performing the inverse operation, which is adding 9 to both sides of the equation. Isolating the variable term is a critical step in solving for x. It involves manipulating the equation to get the term containing x by itself on one side. This process often requires using inverse operations, which are operations that undo each other. In this case, adding 9 is the inverse operation of subtracting 9. By adding 9 to both sides, we maintain the equality of the equation while effectively moving the constant term to the other side. This step is a cornerstone of algebraic problem-solving and is used extensively in various mathematical contexts. It's a fundamental technique for solving equations and inequalities. Without isolating the variable term, it would be impossible to determine the value of x. This step brings us one step closer to the final solution. Remember, the goal is to get 5x by itself, which will then allow us to solve for x directly. Let's proceed with adding 9 to both sides and see how it transforms our equation. This is a crucial step that sets the stage for the final operation of dividing to solve for x.

Adding 9 to both sides of the equation, we get:

5x - 9 + 9 = -29 + 9

5x = -20

We've now simplified the equation to 5x = -20. The final step is to solve for x. To do this, we need to isolate x by dividing both sides of the equation by the coefficient of x, which is 5. This is another application of inverse operations, as division is the inverse of multiplication. Solving for x is the ultimate goal of this process. It involves isolating x completely to determine its value. This is achieved by performing the inverse operation of multiplication, which is division. Dividing both sides of the equation by the coefficient of x ensures that we maintain the equality while effectively isolating x. This is a fundamental technique in algebra and is used extensively in solving equations of various types. The ability to solve for variables is a crucial skill in mathematics and other fields. This final step is the culmination of all the previous steps, bringing us to the solution. Remember, the goal is to get x by itself on one side of the equation, which will directly reveal its value. Let's proceed with dividing both sides by 5 and find the value of x. This step will complete the solution process and provide us with the answer to our problem. With this value, we can verify our solution by substituting it back into the original equation.

Dividing both sides by 5, we get:

(5x) / 5 = -20 / 5

x = -4

In conclusion, we have successfully solved for x in the equation 3(x-3) + 2x = -29. By following the steps of applying the distributive property, combining like terms, isolating the variable term, and dividing to solve for x, we found that x = -4. This process demonstrates the fundamental principles of algebra and how they can be applied to solve linear equations. Solving linear equations is a crucial skill in mathematics and has wide-ranging applications in various fields. The step-by-step approach we've outlined here provides a clear and concise method for tackling similar problems. From understanding the initial equation to arriving at the final solution, each step plays a vital role in the overall process. The distributive property allows us to simplify expressions, combining like terms helps us to condense the equation, isolating the variable term brings us closer to the solution, and dividing completes the process by revealing the value of x. By mastering these techniques, you'll be well-equipped to solve a wide range of algebraic problems. Remember to always check your solution by substituting it back into the original equation to ensure its validity. This verification step is a good practice that helps to prevent errors and build confidence in your problem-solving abilities. As you continue to practice and apply these skills, you'll become more proficient in algebra and gain a deeper understanding of mathematical concepts. This solution not only provides the answer but also reinforces the importance of each step in the process. With a solid foundation in these techniques, you'll be well-prepared to tackle more complex equations and mathematical challenges.

Therefore, the solution to the equation 3(x-3) + 2x = -29 is x = -4.