Solving -11x - 30 + 6x = 0 Find The Value Of X
Linear equations are the backbone of algebra and form the foundation for more advanced mathematical concepts. Understanding how to solve them is crucial for anyone delving into mathematics, physics, engineering, or any field that relies on quantitative analysis. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The most common form of a linear equation is ax + b = 0, where x is the variable, and a and b are constants. The goal of solving a linear equation is to find the value of the variable that makes the equation true. This often involves isolating the variable on one side of the equation by performing the same operations on both sides.
Solving linear equations is not just a theoretical exercise; it has practical applications in various real-world scenarios. For example, you might use a linear equation to calculate how much time it will take to travel a certain distance at a constant speed, or to determine the amount of ingredients needed to scale a recipe. In business, linear equations can help in calculating costs, revenues, and profits. In science, they are used to model relationships between different variables, such as temperature and pressure. The ability to manipulate and solve linear equations is therefore a valuable skill in many areas of life.
Before we dive into the specifics of solving the equation -11x - 30 + 6x = 0, let’s briefly touch upon the fundamental principles that govern the process. The key is to maintain the balance of the equation. Whatever operation you perform on one side, you must perform the same operation on the other side. This ensures that the equality remains valid. Common operations include adding or subtracting the same value from both sides, multiplying or dividing both sides by the same non-zero value, and simplifying expressions by combining like terms. Mastering these basic principles is essential for tackling more complex equations and problems in the future.
Our primary objective is to determine the value of x that satisfies the equation -11x - 30 + 6x = 0. This is a typical linear equation problem that requires us to isolate the variable x on one side of the equation. To achieve this, we will follow a systematic approach, combining like terms and performing algebraic manipulations to simplify the equation. The given equation presents a straightforward scenario, but it is essential to understand each step involved in the solution process to handle more complex equations effectively.
Before we delve into the step-by-step solution, it's important to recognize the different components of the equation. We have two terms involving x (-11x and 6x), a constant term (-30), and the equality sign (=) indicating that the expression on the left-hand side is equal to zero. The presence of multiple x terms means that our first step will likely involve combining these terms to simplify the equation. This is a common strategy in solving linear equations, as it reduces the number of terms and makes the equation easier to manipulate.
To successfully solve this equation, we will employ the fundamental principles of algebra, such as the commutative, associative, and distributive properties. These properties allow us to rearrange and combine terms in a way that helps isolate the variable. The commutative property allows us to change the order of terms without affecting the result (e.g., a + b = b + a). The associative property allows us to group terms differently without changing the outcome (e.g., (a + b) + c = a + (b + c)). The distributive property allows us to multiply a factor across a sum or difference (e.g., a(b + c) = ab + ac). By applying these properties judiciously, we can simplify the equation and make it easier to solve for x.
The first crucial step in solving the equation -11x - 30 + 6x = 0 is to combine the like terms. In this case, the like terms are the terms involving x: -11x and 6x. Combining these terms involves adding their coefficients. The coefficient of a term is the number that is multiplied by the variable. Here, the coefficients are -11 and 6. Adding these coefficients gives us -11 + 6 = -5. Therefore, when we combine the like terms, we get -5x. This simplification reduces the equation to a more manageable form: -5x - 30 = 0.
Now that we have combined the like terms, the equation is -5x - 30 = 0. The next step is to isolate the term containing x. This is achieved by adding 30 to both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance. Adding 30 to both sides gives us -5x - 30 + 30 = 0 + 30. On the left-hand side, the -30 and +30 cancel each other out, leaving us with -5x. On the right-hand side, 0 + 30 equals 30. So, the equation simplifies to -5x = 30.
The final step in solving for x is to divide both sides of the equation by the coefficient of x, which in this case is -5. Dividing both sides by -5 gives us (-5x) / -5 = 30 / -5. On the left-hand side, the -5 in the numerator and the -5 in the denominator cancel each other out, leaving us with x. On the right-hand side, 30 divided by -5 equals -6. Therefore, the solution to the equation is x = -6. This value of x makes the original equation true, meaning that when we substitute -6 for x in the equation -11x - 30 + 6x = 0, the left-hand side will equal zero.
Let's walk through the detailed calculation to ensure a clear understanding of each step. The original equation is -11x - 30 + 6x = 0. Our goal is to isolate x and find its value. The first step, as discussed, is to combine like terms. We identify the terms containing x, which are -11x and 6x. Combining these gives us (-11 + 6)x = -5x. So, the equation becomes -5x - 30 = 0.
Next, we need to isolate the term with x. To do this, we add 30 to both sides of the equation: -5x - 30 + 30 = 0 + 30. This simplifies to -5x = 30. The -30 on the left-hand side is canceled out by the +30, and the 0 on the right-hand side, when added to 30, simply gives us 30. This step is crucial because it moves us closer to isolating x by removing the constant term from the left side of the equation.
Finally, to solve for x, we divide both sides of the equation by the coefficient of x, which is -5: (-5x) / -5 = 30 / -5. On the left-hand side, the -5 in the numerator and the -5 in the denominator cancel out, leaving us with x. On the right-hand side, 30 divided by -5 is -6. Therefore, x = -6. This is the final value of x that satisfies the original equation. To verify this solution, we can substitute x = -6 back into the original equation and check if the equation holds true.
Verifying the solution is a critical step in the process of solving equations. It ensures that the value we have found for the variable actually satisfies the original equation. This step helps to catch any errors that may have occurred during the solution process. To verify the solution x = -6 for the equation -11x - 30 + 6x = 0, we substitute -6 for x in the original equation and check if both sides of the equation are equal.
Substituting x = -6 into the equation -11x - 30 + 6x = 0 gives us -11(-6) - 30 + 6(-6) = 0. Now, we perform the multiplications: -11(-6) = 66 and 6(-6) = -36. So, the equation becomes 66 - 30 - 36 = 0. Next, we perform the subtractions from left to right: 66 - 30 = 36, and then 36 - 36 = 0. Thus, the left-hand side of the equation simplifies to 0, which is equal to the right-hand side of the equation. This confirms that our solution x = -6 is correct.
The verification process not only confirms the correctness of the solution but also reinforces the understanding of the equation and the steps taken to solve it. By substituting the solution back into the original equation, we are essentially reversing the steps we took to isolate the variable. If the equation holds true, it provides a strong indication that our solution is accurate. This practice is particularly useful in more complex problems where the chances of making a mistake are higher. Always remember to verify your solutions to ensure accuracy and confidence in your results.
Having solved the equation -11x - 30 + 6x = 0 and verified the solution, we now need to select the correct answer from the given options. The options provided are:
A. x = 30/17 B. x = -30/17 C. x = 6 D. x = -6
Our calculated solution is x = -6. Comparing this with the given options, we can see that option D, x = -6, matches our solution. Therefore, option D is the correct answer. The other options (A, B, and C) represent incorrect values for x. Option A and B involve fractions, which are not the correct solution for this equation. Option C is the positive counterpart of the correct solution, which is also incorrect.
To further solidify our understanding, let's briefly explain why the other options are incorrect. If we were to substitute x = 30/17 or x = -30/17 into the original equation, the left-hand side would not equal zero. Similarly, if we substituted x = 6, the equation would not hold true. Only when x = -6 does the equation -11x - 30 + 6x = 0 balance, making it the unique solution to the equation. This exercise of comparing the correct solution with the incorrect options highlights the importance of accurate calculations and verification in problem-solving.
When solving linear equations, there are several common mistakes that students often make. Being aware of these mistakes can help you avoid them and improve your problem-solving accuracy. One of the most frequent errors is incorrectly combining like terms. For example, in the equation -11x - 30 + 6x = 0, a common mistake is to add -11x and -30 or to mishandle the signs when combining -11x and 6x. Remember, only terms with the same variable can be combined directly. Ensure you pay close attention to the signs (positive or negative) of the coefficients when adding or subtracting terms.
Another common mistake is not performing the same operation on both sides of the equation. The fundamental principle of solving equations is to maintain balance. If you add, subtract, multiply, or divide on one side, you must do the same on the other side. For instance, in the step where we added 30 to both sides of the equation -5x - 30 = 0, failing to add 30 to both sides would lead to an incorrect result. Similarly, when dividing by the coefficient of x, it's crucial to divide both sides by the same value. Forgetting this principle can throw off the entire solution process.
Sign errors are also a common pitfall in algebra. A simple sign mistake can lead to a completely different answer. In the given equation, errors might occur when multiplying or dividing by negative numbers. For example, dividing 30 by -5 should result in -6, not 6. Double-checking your signs at each step can help prevent these errors. Furthermore, failing to verify the solution is a mistake that many students make. Verification is a crucial step in ensuring the accuracy of your solution. By substituting your solution back into the original equation, you can catch any errors made during the solving process. Always make it a habit to verify your solutions, especially in exams or important assignments.
Solving linear equations can become easier and more efficient with the application of certain tips and tricks. One useful strategy is to simplify the equation as much as possible before attempting to isolate the variable. This often involves combining like terms, distributing terms across parentheses, and clearing fractions or decimals. For example, if an equation contains fractions, multiplying both sides by the least common multiple of the denominators can eliminate the fractions, making the equation easier to work with. Similarly, if an equation contains decimals, multiplying by a power of 10 can clear the decimals.
Another helpful tip is to keep your work organized and write down each step clearly. This not only helps you keep track of your progress but also makes it easier to identify and correct any mistakes. Use a vertical format, aligning the equality signs and keeping like terms in the same column. This visual organization can prevent errors and make the solution process more transparent. Additionally, consider using parentheses to group terms and avoid sign errors, especially when dealing with negative numbers or multiple operations.
Mastering the art of mental math can also significantly speed up your equation-solving abilities. Practice performing simple calculations in your head, such as adding and subtracting integers, multiplying and dividing by small numbers, and simplifying fractions. This can reduce the reliance on calculators and make the process more intuitive. Finally, remember to practice regularly. The more you practice solving linear equations, the more comfortable and confident you will become. Work through a variety of problems, from simple to complex, and challenge yourself to find different approaches and strategies. With consistent practice, you can develop a strong foundation in algebra and excel in more advanced mathematical topics.
In conclusion, solving the linear equation -11x - 30 + 6x = 0 is a fundamental exercise in algebra that demonstrates the importance of understanding and applying basic algebraic principles. By systematically combining like terms, isolating the variable, and performing inverse operations, we arrived at the solution x = -6. This solution was then verified by substituting it back into the original equation, confirming its correctness. The process highlights the significance of each step, from simplifying the equation to avoiding common mistakes and verifying the final answer.
Linear equations are not just abstract mathematical constructs; they have wide-ranging applications in various fields, including science, engineering, economics, and everyday problem-solving. The ability to confidently solve linear equations is a valuable skill that forms the basis for more advanced mathematical concepts and real-world applications. Mastering these equations requires a solid understanding of the underlying principles, careful attention to detail, and consistent practice.
Remember, solving equations is not just about finding the right answer; it's about developing a logical and systematic approach to problem-solving. By understanding the steps involved, avoiding common mistakes, and practicing regularly, you can build a strong foundation in algebra and enhance your mathematical abilities. Keep practicing, keep learning, and you will find that solving linear equations becomes second nature. The journey through algebra is a rewarding one, and the skills you acquire will serve you well in many aspects of life and career.
