Solving For X In 2x - 3x - 18 = -18 + 2x + 5x A Step-by-Step Guide

Introduction

This article provides a comprehensive guide on how to solve for x in the equation 2x - 3x - 18 = -18 + 2x + 5x. We will break down the steps involved in solving this linear equation, explain the underlying principles, and discuss common mistakes to avoid. This guide is designed to be accessible to anyone, regardless of their mathematical background. Whether you're a student learning algebra or simply want to brush up on your skills, this article will provide you with the knowledge and confidence to tackle similar problems. Understanding how to solve equations like this is a fundamental skill in algebra and has wide-ranging applications in various fields, including science, engineering, and finance. We will explore the solution process in detail, ensuring that you grasp not only the steps but also the reasoning behind them. So, let's dive in and master the art of solving linear equations!

Understanding Linear Equations

Before we jump into solving the equation, let's first understand what a linear equation is. 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 variable is raised to the power of one, and there are no exponents or other complex functions involved. The general form of a linear equation is ax + b = 0, where a and b are constants and x is the variable. In our equation, 2x - 3x - 18 = -18 + 2x + 5x, we can see that it fits this definition. There are terms with x (2x, -3x, 2x, 5x) and constant terms (-18, -18). Solving a linear equation means finding the value of the variable (x in this case) that makes the equation true. This involves manipulating the equation using algebraic operations to isolate the variable on one side. These operations include addition, subtraction, multiplication, and division, and they must be applied to both sides of the equation to maintain equality. Understanding the properties of equality is crucial for solving linear equations accurately. For example, the addition property of equality states that adding the same quantity to both sides of an equation does not change the solution. Similarly, the multiplication property of equality states that multiplying both sides of an equation by the same non-zero quantity does not change the solution. By applying these properties systematically, we can simplify the equation and ultimately find the value of x.

Step-by-Step Solution

Now, let's solve the equation 2x - 3x - 18 = -18 + 2x + 5x step-by-step. This step-by-step solution will guide you through the process of isolating x and finding its value.

Step 1: Combine Like Terms on Each Side

The first step is to simplify both sides of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. On the left side, we have 2x and -3x, which can be combined to give -x. So, the left side simplifies to -x - 18. On the right side, we have 2x and 5x, which combine to give 7x. Thus, the right side simplifies to -18 + 7x. Our equation now looks like this: -x - 18 = -18 + 7x. Combining like terms is a crucial step in simplifying equations, as it reduces the number of terms and makes the equation easier to manipulate. This step relies on the distributive property and the commutative property of addition. By combining like terms, we are essentially grouping the terms with the same variable together and the constant terms together. This makes it easier to isolate the variable in subsequent steps. It is important to pay attention to the signs of the terms when combining them, as incorrect signs can lead to errors in the solution.

Step 2: Isolate the Variable Terms on One Side

Next, we want to isolate the variable terms on one side of the equation and the constant terms on the other side. To do this, we can add x to both sides of the equation. Adding x to both sides cancels out the -x on the left side, leaving us with -18 = -18 + 8x. Now, we have all the variable terms on the right side. Alternatively, we could have subtracted 7x from both sides, which would have resulted in all the variable terms being on the left side. The choice of which side to isolate the variable terms on is often a matter of personal preference, but it is important to choose a method and stick with it throughout the solution process. The goal is to simplify the equation in a way that makes it easier to isolate the variable. By isolating the variable terms on one side, we are effectively grouping all the terms with x together, which is a necessary step in solving for x. This step utilizes the addition property of equality, which states that adding the same quantity to both sides of an equation does not change the solution.

Step 3: Isolate the Constant Terms on the Other Side

Now, we need to isolate the constant terms on the other side of the equation. We can do this by adding 18 to both sides. Adding 18 to both sides cancels out the -18 on both sides, resulting in 0 = 8x. We now have the equation in a very simplified form, with the variable term on one side and the constant term on the other. Isolating the constant terms is the counterpart to isolating the variable terms. By grouping all the constant terms together, we are making it easier to see the relationship between the variable and the constant. This step also utilizes the addition property of equality. Adding 18 to both sides ensures that we maintain the balance of the equation while moving the constant terms to the right side. The resulting equation, 0 = 8x, is a clear indication of the next step required to solve for x.

Step 4: Solve for x

Finally, to solve for x, we need to divide both sides of the equation by the coefficient of x, which is 8. Dividing both sides of 0 = 8x by 8 gives us x = 0. Therefore, the solution to the equation 2x - 3x - 18 = -18 + 2x + 5x is x = 0. This is the final step in the solution process. Dividing both sides by the coefficient of x isolates x and gives us its value. This step utilizes the division property of equality, which states that dividing both sides of an equation by the same non-zero quantity does not change the solution. In this case, dividing both sides by 8 gives us the value of x directly. The solution x = 0 means that when we substitute 0 for x in the original equation, the equation holds true. This can be verified by substituting x = 0 into the original equation and checking if both sides are equal.

Verification of the Solution

To ensure that our solution is correct, it's always a good practice to verify it. We can do this by substituting x = 0 back into the original equation: 2(0) - 3(0) - 18 = -18 + 2(0) + 5(0). Simplifying both sides, we get -18 = -18, which is true. This confirms that our solution x = 0 is correct. Verifying the solution is a crucial step in the problem-solving process. It helps to catch any errors that may have occurred during the solution process and ensures that the final answer is correct. By substituting the solution back into the original equation, we are essentially checking if the value of x makes the equation a true statement. If both sides of the equation are equal after the substitution, then the solution is correct. If the two sides are not equal, then there is an error in the solution process, and we need to go back and check each step to identify the mistake. Verification provides confidence in the solution and reinforces the understanding of the equation and its properties. In this case, the verification confirms that x = 0 is indeed the correct solution.

Common Mistakes to Avoid

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 accuracy. One common mistake is not distributing correctly when dealing with parentheses. For example, if the equation had terms like 2(x + 3), you need to multiply both x and 3 by 2. Another common mistake is combining unlike terms. Remember, you can only combine terms that have the same variable raised to the same power. For example, you cannot combine 2x and 2x². Another frequent error is forgetting to apply the same operation to both sides of the equation. The properties of equality require that any operation performed on one side of the equation must also be performed on the other side to maintain balance. Sign errors are also common, especially when dealing with negative numbers. Pay close attention to the signs of the terms when combining like terms or moving terms from one side of the equation to the other. Finally, it's important to verify your solution by substituting it back into the original equation. This will help you catch any mistakes you may have made during the solution process. By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your ability to solve linear equations accurately and efficiently. Practicing regularly and reviewing your work can also help reinforce your understanding and reduce the likelihood of making errors.

Conclusion

In conclusion, solving for x in the equation 2x - 3x - 18 = -18 + 2x + 5x involves a series of algebraic manipulations to isolate the variable. By following the steps outlined in this guide – combining like terms, isolating variable terms, isolating constant terms, and solving for x – we have found that the solution is x = 0. Remember to always verify your solution to ensure accuracy. Understanding the principles behind solving linear equations is essential for success in algebra and beyond. Linear equations are the foundation for more advanced mathematical concepts, and mastering them will open doors to a wide range of applications in various fields. By practicing regularly and paying attention to detail, you can develop the skills and confidence needed to tackle any linear equation. The key is to break down the problem into smaller, manageable steps and apply the properties of equality systematically. With consistent effort and a solid understanding of the fundamentals, you can become proficient in solving linear equations and use this knowledge to solve more complex problems. So, keep practicing, keep learning, and keep exploring the fascinating world of mathematics!

Final Answer: A. x=0