Solving Linear Equations A Step-by-Step Guide To 4(x-2) - 3(x-1) = 2(x+6)
Introduction to Solving Linear Equations
In the realm of mathematics, solving equations is a fundamental skill that underpins numerous concepts and applications. Linear equations, in particular, form the bedrock of algebraic problem-solving. They appear in various forms, from simple expressions to complex formulations, and mastering their resolution is crucial for mathematical proficiency. This article delves into the step-by-step process of solving a specific linear equation: 4(x-2) - 3(x-1) = 2(x+6). We will explore the underlying principles, demonstrate the algebraic manipulations involved, and provide a clear, concise solution. Understanding the methodology presented here will not only empower you to solve similar equations but also enhance your overall mathematical reasoning.
The beauty of linear equations lies in their straightforward structure and the systematic approach to finding their solutions. Unlike more complex equations that might require advanced techniques, linear equations can be tackled using basic algebraic operations. The key is to isolate the variable (in this case, 'x') on one side of the equation, revealing its value. This process often involves expanding expressions, combining like terms, and applying the properties of equality. By carefully executing these steps, we can unravel the solution and gain a deeper appreciation for the elegance of mathematical problem-solving.
Before we dive into the specifics of our equation, it's important to grasp the core concepts that govern linear equations. A linear equation is essentially an algebraic statement that asserts the equality of two expressions. These expressions involve a variable raised to the power of one (hence the term "linear") and can include constants and coefficients. The goal of solving a linear equation is to find the value(s) of the variable that make the equation true. In other words, we seek the value(s) that, when substituted into the equation, will make both sides of the equation equal. This quest for the variable's value is what drives the entire solution process.
Step-by-Step Solution: 4(x-2) - 3(x-1) = 2(x+6)
1. Expanding the Expressions
The first step in solving the equation 4(x-2) - 3(x-1) = 2(x+6) is to expand the expressions on both sides of the equation. This involves applying the distributive property, which states that a(b + c) = ab + ac. By distributing the constants outside the parentheses, we can eliminate the parentheses and simplify the equation. This is a crucial step because it allows us to combine like terms and move closer to isolating the variable 'x'. Neglecting this initial expansion can lead to errors in subsequent steps, making it essential to master this fundamental algebraic technique.
Let's begin by expanding the left side of the equation. We have 4(x-2), which can be expanded as 4 * x - 4 * 2, resulting in 4x - 8. Similarly, -3(x-1) expands to -3 * x + (-3) * (-1), which gives us -3x + 3. Combining these expanded terms, the left side of the equation becomes 4x - 8 - 3x + 3. It's crucial to pay attention to the signs when distributing, as a sign error can significantly alter the solution.
Next, we expand the right side of the equation, 2(x+6). Applying the distributive property, we get 2 * x + 2 * 6, which simplifies to 2x + 12. Now, we have successfully expanded both sides of the equation, transforming it into a more manageable form. The expanded equation is 4x - 8 - 3x + 3 = 2x + 12. This form is easier to work with because it allows us to identify and combine like terms, a necessary step towards isolating the variable 'x'.
The expansion process is not just about applying the distributive property; it's also about careful attention to detail and a methodical approach. Each term must be correctly multiplied, and the signs must be accurately tracked. This meticulousness is a hallmark of effective algebraic manipulation and a key to avoiding common errors. By expanding the expressions in a clear and systematic manner, we set the stage for the subsequent steps in the solution process.
2. Combining Like Terms
After expanding the expressions, the next essential step in solving the equation 4(x-2) - 3(x-1) = 2(x+6) is to combine like terms on each side of the equation. Like terms are terms that contain the same variable raised to the same power. In our equation, terms with 'x' and constant terms can be combined separately. This simplification process makes the equation more concise and easier to manipulate, bringing us closer to isolating the variable 'x'.
On the left side of the equation, we have the terms 4x and -3x, which are like terms. Combining them, we get 4x - 3x = x. We also have the constant terms -8 and +3. Combining these, we get -8 + 3 = -5. Thus, the left side of the equation simplifies to x - 5. This combination significantly reduces the complexity of the expression, making it more manageable for the next steps.
On the right side of the equation, we have 2x + 12. There are no like terms to combine on this side, so it remains as 2x + 12. Now, the equation has been simplified to x - 5 = 2x + 12. This form is much cleaner and easier to work with than the original equation. The process of combining like terms is a critical step in solving linear equations because it streamlines the equation and reduces the potential for errors in later stages.
The ability to identify and combine like terms is a fundamental skill in algebra. It requires a clear understanding of what constitutes a like term and careful attention to the signs of the terms. By methodically combining like terms, we simplify the equation and make it more amenable to further algebraic manipulation. This step is not just about reducing complexity; it's about making the equation more transparent and revealing the underlying relationships between the variables and constants.
3. Isolating the Variable
Isolating the variable is the heart of solving any linear equation, including 4(x-2) - 3(x-1) = 2(x+6). After expanding and combining like terms, we've simplified the equation to x - 5 = 2x + 12. Now, our goal is to get 'x' by itself on one side of the equation. This involves using inverse operations to move terms around while maintaining the equality of the equation. The key principle here is that whatever operation we perform on one side of the equation, we must perform on the other side to keep the equation balanced.
To isolate 'x', let's first eliminate the 'x' term on the right side of the equation. We can do this by subtracting 2x from both sides. This gives us x - 5 - 2x = 2x + 12 - 2x. Simplifying, we get -x - 5 = 12. This step moves all the 'x' terms to the left side of the equation, which is a crucial step towards isolating 'x'.
Next, we need to eliminate the constant term on the left side of the equation. We have -x - 5 = 12. To eliminate the -5, we add 5 to both sides of the equation: -x - 5 + 5 = 12 + 5. This simplifies to -x = 17. We're almost there, but we still have a negative sign on the 'x'.
Finally, to get 'x' by itself, we multiply both sides of the equation by -1. This gives us (-1) * (-x) = (-1) * 17, which simplifies to x = -17. We have successfully isolated the variable 'x' and found its value. This process of using inverse operations to move terms and isolate the variable is a fundamental skill in algebra and is applicable to a wide range of equations.
4. Verifying the Solution
After finding a solution to an equation, it's crucial to verify that the solution is correct. This step ensures that we haven't made any errors in our algebraic manipulations and that the value we found for the variable actually satisfies the original equation. For the equation 4(x-2) - 3(x-1) = 2(x+6), we found the solution to be x = -17. To verify this, we substitute -17 for 'x' in the original equation and check if both sides of the equation are equal.
Substituting x = -17 into the left side of the equation, we get 4((-17)-2) - 3((-17)-1). Simplifying inside the parentheses, we have 4(-19) - 3(-18). Multiplying, we get -76 + 54, which equals -22. So, the left side of the equation evaluates to -22 when x = -17.
Now, we substitute x = -17 into the right side of the equation, 2(x+6). This gives us 2((-17)+6). Simplifying inside the parentheses, we have 2(-11). Multiplying, we get -22. So, the right side of the equation also evaluates to -22 when x = -17.
Since both the left side and the right side of the equation equal -22 when x = -17, we have verified that our solution is correct. This verification step is an essential part of the problem-solving process, as it provides confidence in our answer and ensures that we haven't made any algebraic errors. By substituting the solution back into the original equation, we can confirm its validity and gain a deeper understanding of the equation's behavior.
Conclusion
In conclusion, solving the linear equation 4(x-2) - 3(x-1) = 2(x+6) involves a systematic approach that includes expanding expressions, combining like terms, isolating the variable, and verifying the solution. We have demonstrated each of these steps in detail, providing a clear pathway to finding the solution x = -17. This process not only solves the specific equation but also reinforces fundamental algebraic principles that are essential for mathematical proficiency.
The ability to solve linear equations is a cornerstone of mathematical understanding. It's a skill that is applied in various contexts, from simple problem-solving to complex scientific and engineering applications. By mastering the techniques presented in this article, you will be well-equipped to tackle a wide range of mathematical challenges. Remember, the key to success in mathematics is not just about finding the right answer; it's about understanding the underlying principles and developing a systematic approach to problem-solving.
The steps we've outlined – expanding, combining, isolating, and verifying – are not just specific to this equation; they are a general framework for solving linear equations. By internalizing this framework, you can approach any linear equation with confidence and clarity. Mathematics is a journey of discovery, and each equation you solve is a step forward in your understanding of the mathematical world. Keep practicing, keep exploring, and keep solving!
