Solving The Linear Equation -2(2x-11)=3(x-9) A Step-by-Step Guide
In this comprehensive guide, we will delve into the process of solving linear equations, using the example -2(2x - 11) = 3(x - 9) to illustrate the step-by-step approach. Mastering the art of solving linear equations is fundamental in mathematics, serving as a cornerstone for more advanced algebraic concepts. This article aims to provide a clear, concise, and thorough explanation, making it accessible to learners of all levels. We'll not only break down the solution to this specific equation but also highlight the underlying principles and techniques applicable to a wide range of linear equations. By the end of this guide, you'll be equipped with the skills and confidence to tackle similar problems with ease.
Understanding the Basics of Linear Equations
Before we dive into the solution, let's establish a firm understanding of what linear equations are and the basic principles involved in solving them. 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. These equations are called “linear” because when plotted on a graph, they form a straight line. The goal in solving a linear equation is to isolate the variable (in this case, x) on one side of the equation to determine its value. This involves using algebraic manipulations to maintain the equality while simplifying the equation.
The fundamental principle in solving any equation is maintaining balance. Think of the equation as a weighing scale; whatever operation you perform on one side, you must perform on the other to keep the scale balanced. This principle is crucial as we apply various operations to both sides of our equation to isolate x. These operations typically include addition, subtraction, multiplication, and division. The key is to perform these operations strategically, aiming to simplify the equation step by step until we arrive at the solution. Furthermore, understanding the order of operations (PEMDAS/BODMAS) is vital in correctly simplifying expressions within the equation. This ensures that we handle parentheses, exponents, multiplication, division, addition, and subtraction in the correct sequence, leading to an accurate solution. In the following sections, we will apply these principles to solve the equation -2(2x - 11) = 3(x - 9), demonstrating each step in detail.
Step-by-Step Solution to -2(2x - 11) = 3(x - 9)
Now, let's embark on the journey of solving the equation -2(2x - 11) = 3(x - 9). We'll dissect each step, providing a clear rationale for every action taken. This methodical approach will not only help you understand the solution to this specific problem but also equip you with the skills to tackle similar equations.
Step 1: Distribute the Constants
The first step in simplifying the equation is to distribute the constants outside the parentheses. This involves multiplying the constant by each term inside the parentheses. For the left side of the equation, we have -2 multiplied by (2x - 11). Applying the distributive property, we get:
-2 * 2x = -4x -2 * -11 = 22
So, the left side of the equation becomes -4x + 22. Similarly, for the right side of the equation, we have 3 multiplied by (x - 9). Distributing the 3, we get:
3 * x = 3x 3 * -9 = -27
Therefore, the right side of the equation becomes 3x - 27. After distributing the constants, our equation now looks like this: -4x + 22 = 3x - 27. This step is crucial as it removes the parentheses, making it easier to combine like terms and isolate the variable.
Step 2: Combine Like Terms
After distributing the constants, the next step is to combine like terms. This involves grouping terms with the variable x on one side of the equation and the constant terms on the other side. To achieve this, we'll perform algebraic manipulations to move terms across the equals sign. Remember, whatever operation we perform on one side, we must perform on the other to maintain the balance of the equation.
Let's start by moving the term with x from the right side to the left side. We have 3x on the right side, so we subtract 3x from both sides of the equation:
-4x + 22 - 3x = 3x - 27 - 3x
This simplifies to:
-7x + 22 = -27
Now, we need to move the constant term from the left side to the right side. We have 22 on the left side, so we subtract 22 from both sides of the equation:
-7x + 22 - 22 = -27 - 22
This simplifies to:
-7x = -49
At this point, all the x terms are on the left side, and all the constant terms are on the right side. This sets us up for the final step of isolating x.
Step 3: Isolate the Variable
The final step in solving for x is to isolate the variable. We have -7x = -49, which means -7 multiplied by x equals -49. To isolate x, we need to undo the multiplication by -7. This is achieved by dividing both sides of the equation by -7:
-7x / -7 = -49 / -7
When we divide -7x by -7, we are left with x. On the right side, -49 divided by -7 equals 7. Therefore, the equation simplifies to:
x = 7
This is our solution. We have successfully isolated x and found its value to be 7. To ensure our solution is correct, we can substitute x = 7 back into the original equation and verify that both sides are equal.
Verification of the Solution
To ensure the accuracy of our solution, it's crucial to verify the solution by substituting the value of x back into the original equation. This process confirms that our calculated value satisfies the equation, providing us with confidence in our answer. Let's substitute x = 7 into the original equation, -2(2x - 11) = 3(x - 9):
-2(2 * 7 - 11) = 3(7 - 9)
First, we simplify the expressions inside the parentheses:
-2(14 - 11) = 3(-2)
-2(3) = 3(-2)
Now, we perform the multiplication:
-6 = -6
As we can see, both sides of the equation are equal when x = 7. This confirms that our solution is correct. Verification is a vital step in problem-solving, especially in mathematics. It helps identify any potential errors in the solution process and ensures that the final answer is accurate. In this case, our verification confirms that x = 7 is indeed the correct solution to the equation -2(2x - 11) = 3(x - 9).
Common Mistakes to Avoid
Solving linear equations can sometimes be tricky, and it's easy to make mistakes if you're not careful. Understanding common mistakes and how to avoid them is crucial for mastering this fundamental mathematical skill. One frequent error is incorrect distribution. When multiplying a constant by an expression in parentheses, ensure that you multiply the constant by each term inside the parentheses. For example, in the equation -2(2x - 11) = 3(x - 9), a mistake would be to multiply -2 only by 2x, forgetting to multiply it by -11. This can lead to an incorrect equation and subsequently, a wrong solution.
Another common mistake is combining unlike terms. Remember, you can only add or subtract terms that have the same variable and exponent. For instance, you can combine -4x and -3x because they both have the variable x raised to the power of 1. However, you cannot combine -4x with 22 because 22 is a constant term without a variable. Mixing up like and unlike terms can result in a simplified equation that doesn't accurately represent the original problem.
Sign errors are also a common pitfall. Pay close attention to the signs (positive or negative) of the terms when performing operations. A simple sign mistake can completely alter the solution. For example, when moving terms from one side of the equation to the other, remember to change their signs. If you have +3x on the right side and move it to the left side, it becomes -3x. Neglecting this sign change can lead to an incorrect balance of the equation.
Finally, forgetting to perform the same operation on both sides of the equation is a critical error. The principle of maintaining balance is fundamental to solving equations. If you add, subtract, multiply, or divide on one side, you must do the same on the other side. Failing to do so disrupts the equality and leads to an incorrect solution. By being mindful of these common mistakes and practicing careful, step-by-step solutions, you can significantly improve your accuracy in solving linear equations.
Conclusion: Mastering Linear Equations
In conclusion, mastering the skill of solving linear equations is a fundamental building block in mathematics. Throughout this guide, we've meticulously dissected the process of solving the equation -2(2x - 11) = 3(x - 9), providing a step-by-step approach that demystifies the process. We began by understanding the basics of linear equations, emphasizing the importance of maintaining balance while performing algebraic manipulations. We then moved on to the actual solution, distributing constants, combining like terms, and isolating the variable x. Our solution revealed that x = 7.
To reinforce the accuracy of our solution, we undertook a verification step, substituting x = 7 back into the original equation and confirming that both sides were indeed equal. This verification process not only validates our answer but also underscores the importance of double-checking our work to prevent errors. Furthermore, we highlighted common mistakes to avoid, such as incorrect distribution, combining unlike terms, sign errors, and neglecting to perform the same operation on both sides of the equation. By being aware of these pitfalls, you can enhance your problem-solving skills and minimize the chances of making errors.
Solving linear equations is not just about finding the correct answer; it's about developing a logical and systematic approach to problem-solving. The techniques learned here can be applied to a wide range of mathematical problems and are essential for further studies in algebra and beyond. Practice is key to mastering these skills. The more you practice, the more confident and proficient you will become in solving linear equations. So, take the time to work through various examples, apply the steps we've discussed, and always verify your solutions. With consistent effort, you'll be well-equipped to tackle any linear equation that comes your way.
