Solve For X In The Equation 4x - 7 - 2x = 11.
In the realm of algebra, simplifying expressions and solving equations are fundamental skills. One crucial technique is combining like terms, which allows us to condense expressions and make them easier to manipulate. This skill is particularly essential when solving equations, as it helps us isolate the variable and find its value. In this guide, we will delve into the process of combining like terms and then apply this technique to solve a linear equation for the variable x. Specifically, we will explore the equation 4x - 7 - 2x = 11, meticulously walking through each step to arrive at the solution. Understanding how to combine like terms is not just about finding the right answer; it's about building a solid foundation for more advanced algebraic concepts. Let's embark on this journey of mathematical discovery and unravel the intricacies of solving for x.
Understanding Like Terms
Before we dive into the equation itself, let's first define what like terms are. Like terms are terms that have the same variable raised to the same power. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical. For example, 4x and -2x are like terms because they both have the variable x raised to the power of 1. On the other hand, 4x and 4x² are not like terms because the variable x is raised to different powers (1 and 2, respectively). Similarly, 7 and 11 are like terms because they are both constants (numbers without any variables). Identifying like terms is the first step in simplifying expressions. Once we have identified them, we can combine them by adding or subtracting their coefficients, while keeping the variable part the same. This process is crucial for making equations easier to work with and ultimately solving for the unknown variable.
Combining Like Terms in the Equation
Now, let's apply this concept to our equation: 4x - 7 - 2x = 11. Our goal is to combine the like terms on the left side of the equation to simplify it. Looking at the left side, we can identify two terms that contain the variable x: 4x and -2x. These are like terms, so we can combine them. To do this, we simply add their coefficients: 4 + (-2) = 2. This means that 4x - 2x simplifies to 2x. The constant term -7 does not have any other like terms on the left side of the equation, so it remains as it is. After combining like terms, our equation becomes 2x - 7 = 11. Notice how the equation is now simpler and easier to solve. Combining like terms is a fundamental step in solving equations, as it reduces the complexity and allows us to isolate the variable more effectively. This process of simplification is not just about making the equation shorter; it's about revealing the underlying structure and making the solution path clearer.
Isolating the Variable
With our equation simplified to 2x - 7 = 11, the next step is to isolate the variable x. This means getting x by itself on one side of the equation. To do this, we need to undo any operations that are being performed on x. In this case, x is being multiplied by 2 and then 7 is being subtracted. We will undo these operations in reverse order. First, we need to undo the subtraction of 7. To do this, we add 7 to both sides of the equation. Adding the same value to both sides of an equation maintains the equality, which is a fundamental principle in algebra. So, we have 2x - 7 + 7 = 11 + 7. Simplifying this, we get 2x = 18. Now, we have isolated the term with x on one side of the equation. The next step is to undo the multiplication by 2. This will give us the value of x.
Solving for x
To undo the multiplication of 2 in the equation 2x = 18, we need to perform the inverse operation, which is division. We divide both sides of the equation by 2. Again, this maintains the equality because we are performing the same operation on both sides. So, we have (2x) / 2 = 18 / 2. Simplifying this, we get x = 9. This is our solution: the value of x that makes the original equation true. We have successfully isolated x and found its value by combining like terms and performing inverse operations. This process demonstrates the power of algebraic manipulation in solving equations. It's not just about following steps; it's about understanding the underlying principles of equality and inverse operations.
Verification of the Solution
To ensure that our solution is correct, we can verify it by substituting the value of x back into the original equation. This is a crucial step in problem-solving, as it helps us catch any potential errors. Our original equation was 4x - 7 - 2x = 11. We found that x = 9, so we substitute 9 for x in the equation: 4(9) - 7 - 2(9) = 11. Now, we simplify the left side of the equation: 36 - 7 - 18 = 11. Continuing the simplification, we have 29 - 18 = 11, which gives us 11 = 11. Since the left side of the equation equals the right side, our solution is correct. This verification step reinforces the understanding of how the solution satisfies the equation. It also highlights the importance of accuracy in each step of the solving process. By verifying our solution, we can be confident that we have found the correct value for x.
Conclusion
In this comprehensive guide, we have explored the process of combining like terms and solving for x in the equation 4x - 7 - 2x = 11. We began by defining like terms and understanding their importance in simplifying expressions. We then applied this concept to combine like terms in the equation, which led to a simpler form: 2x - 7 = 11. Next, we focused on isolating the variable x by performing inverse operations, adding 7 to both sides and then dividing by 2. This yielded the solution x = 9. Finally, we verified our solution by substituting it back into the original equation, confirming its correctness. This step-by-step approach demonstrates the power of algebraic manipulation in solving equations. Understanding how to combine like terms, isolate variables, and verify solutions is crucial for success in algebra and beyond. These skills are not only applicable to solving equations but also form the foundation for more advanced mathematical concepts. By mastering these techniques, you can confidently tackle a wide range of algebraic problems and develop a deeper understanding of mathematical principles. The journey of learning algebra is a continuous process of building skills and expanding knowledge. Keep practicing, keep exploring, and you will unlock the power of mathematics!