Solving Linear Equations A Step-by-Step Guide With 4(x-8)+2x = 6(x+2)-44

In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a code, where we aim to find the value of an unknown variable that makes the equation true. This article delves into the process of solving a specific linear equation, providing a step-by-step guide that demystifies the process. This specific case focuses on the equation $4(x-8)+2x = 6(x+2)-44$. Linear equations, characterized by variables raised to the power of one, are the building blocks of algebra and find applications in various fields, from everyday calculations to advanced scientific models. Mastering the art of solving linear equations is crucial for anyone venturing into the world of mathematics and its applications. This guide aims to provide a comprehensive understanding of the techniques involved, ensuring clarity and confidence in tackling similar problems.

The initial equation we are tasked with solving is $4(x-8) + 2x = 6(x+2) - 44$. This equation, at first glance, might seem complex, but it's essentially a puzzle waiting to be solved. Our mission is to isolate the variable 'x' on one side of the equation, thereby unveiling its value. To embark on this journey, we'll employ a series of algebraic manipulations, each step bringing us closer to the solution. These manipulations are based on the fundamental principles of equality, ensuring that the balance of the equation is maintained throughout the process. Think of it as a balancing scale, where any operation performed on one side must also be mirrored on the other side to keep the scale level. The steps we'll take involve distributing, combining like terms, and strategically moving terms across the equal sign. Each of these steps is a tool in our mathematical toolbox, and when wielded correctly, they lead us to the coveted solution. Before diving into the specifics, it's crucial to understand the underlying logic. We're not just mindlessly following rules; we're applying principles that guarantee the integrity of the equation. This understanding is what separates rote memorization from true mathematical proficiency. As we progress through the steps, we'll not only focus on the 'how' but also the 'why,' solidifying your grasp of the concepts. So, let's put on our mathematical hats and embark on this exciting quest to solve for 'x'.

The first step in solving this equation involves distributing the constants outside the parentheses. This means multiplying the number outside the parenthesis by each term inside the parenthesis. For the left side of the equation, we have $4(x-8)$, which expands to $4 * x - 4 * 8$, which simplifies to $4x - 32$. Similarly, on the right side, we have $6(x+2)$, which expands to $6 * x + 6 * 2$, which simplifies to $6x + 12$. This distribution step is crucial because it eliminates the parentheses, making the equation easier to manipulate. It's like unwrapping a present to reveal the contents inside. By removing the parentheses, we expose the individual terms, allowing us to combine like terms and move closer to isolating 'x'.

After performing the distribution, the equation transforms from $4(x-8) + 2x = 6(x+2) - 44$ to $4x - 32 + 2x = 6x + 12 - 44$. Notice how the parentheses have vanished, and we're left with a string of individual terms connected by addition and subtraction. This new form is much more manageable. We can now clearly see the 'x' terms and the constant terms, which are the building blocks we'll use to solve for 'x'. The distribution step is not just a mechanical process; it's a strategic move that simplifies the equation and sets the stage for further manipulations. It's like clearing the clutter in a room before you start organizing the furniture. By removing the parentheses, we've created a clear space to work, making the subsequent steps much smoother. It's important to double-check your distribution to ensure accuracy. A small error in this step can cascade through the rest of the solution, leading to a wrong answer. So, take your time, be meticulous, and ensure that the constants are distributed correctly. With the distribution complete, we're ready to move on to the next step, which involves combining like terms.

Now that we've distributed the constants, the next logical step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, $4x - 32 + 2x = 6x + 12 - 44$, the like terms on the left side are $4x$ and $2x$, and the like terms on the right side are $12$ and $-44$. Combining $4x$ and $2x$ gives us $6x$. Combining $12$ and $-44$ gives us $-32$. This process of combining like terms is essentially simplifying the equation by grouping together terms that are similar. It's like sorting your socks by color to make it easier to find a matching pair. By combining like terms, we reduce the number of terms in the equation, making it more concise and easier to solve.

After combining like terms, the equation simplifies to $6x - 32 = 6x - 32$. This is a significant simplification from the previous form. Notice how the equation is now much more compact and symmetrical. We've reduced the number of terms, making the equation easier to analyze. The act of combining like terms is not just about making the equation look simpler; it's about revealing the underlying structure of the equation. It allows us to see the relationships between the variables and the constants more clearly. In this particular case, the result of combining like terms reveals something quite interesting: both sides of the equation are identical. This hints at a special type of solution, which we'll explore further in the next steps. But before we jump ahead, it's crucial to ensure that the like terms have been combined correctly. A mistake in this step can lead to an incorrect solution. So, double-check your work, make sure you've identified all the like terms, and that you've combined them accurately. With the like terms combined, we're one step closer to unraveling the mystery of this equation.

At this stage, our equation is $6x - 32 = 6x - 32$. The goal now is to isolate the variable 'x'. This means getting all the 'x' terms on one side of the equation and all the constant terms on the other side. To do this, we can subtract $6x$ from both sides of the equation. This step is based on the principle of equality, which states that we can perform the same operation on both sides of an equation without changing its balance. Subtracting $6x$ from both sides eliminates the 'x' terms from both sides, leaving us with an equation involving only constants.

Performing this subtraction, we get $6x - 32 - 6x = 6x - 32 - 6x$, which simplifies to $-32 = -32$. This result is quite intriguing. The variable 'x' has completely disappeared from the equation, and we're left with a statement that is always true. This indicates that the original equation is an identity, meaning that it's true for all values of 'x'. In other words, no matter what number we substitute for 'x' in the original equation, the equation will hold true. This is a special case in equation solving, and it's important to recognize it when it occurs. It's like discovering a hidden truth within the equation itself. The fact that the variable has vanished and we're left with a true statement is a clear sign that we're dealing with an identity. This outcome is a testament to the power of algebraic manipulation. By systematically applying the rules of algebra, we've uncovered a fundamental property of the equation. It's a reminder that solving equations is not just about finding a single numerical answer; it's about understanding the relationships between variables and constants. With this understanding, we can confidently conclude that the solution to the equation is all real numbers.

Having simplified the equation to $-32 = -32$, we've arrived at a crucial point: interpreting the solution. As we observed, the variable 'x' has vanished, and we're left with a true statement. This indicates that the original equation is an identity. An identity is an equation that is true for all values of the variable. This means that no matter what number we substitute for 'x', the equation will always hold true. Unlike equations with a single solution or no solution, an identity has an infinite number of solutions. Every real number is a solution to this equation.

The significance of this outcome lies in the nature of the equation itself. The fact that it simplifies to a true statement regardless of the value of 'x' reveals a fundamental relationship between the terms. It suggests that the left-hand side and the right-hand side of the original equation are essentially the same, just expressed in different forms. This understanding is crucial for developing a deeper appreciation of algebraic equations. It's not just about finding a numerical answer; it's about recognizing patterns and relationships. The concept of an identity is a powerful one in mathematics. It's used in various areas, from simplifying expressions to proving theorems. Recognizing an identity can save time and effort in problem-solving, as it eliminates the need to search for a specific numerical solution. In our case, the equation $4(x-8) + 2x = 6(x+2) - 44$ is not just a problem to be solved; it's a statement about equality that holds true universally. This is a valuable insight that goes beyond the immediate task of solving the equation. It's a lesson in the beauty and elegance of mathematics, where seemingly complex expressions can reveal hidden truths. Therefore, the solution to this equation is all real numbers, reflecting the identity nature of the equation.

In conclusion, after systematically applying the principles of algebra, we've successfully solved the equation $4(x-8) + 2x = 6(x+2) - 44$. The journey involved distributing constants, combining like terms, and isolating the variable. However, the most significant outcome was the realization that the equation simplifies to $-32 = -32$, a true statement devoid of the variable 'x'. This revelation signifies that the equation is an identity, a special type of equation that holds true for all values of the variable.

Therefore, the solution to the equation is all real numbers. This means that any real number, whether positive, negative, zero, integer, fraction, or irrational number, can be substituted for 'x' in the original equation, and the equation will remain valid. This is a powerful result that underscores the nature of identities in mathematics. Identities are not merely equations with infinite solutions; they are statements of fundamental equivalence. They reveal that two seemingly different expressions are, in fact, identical in their underlying structure. The process of solving this equation has not only provided us with the solution but has also deepened our understanding of algebraic principles. We've seen how distribution, combining like terms, and isolating variables are not just mechanical steps but strategic tools that allow us to unravel the complexities of equations. Furthermore, we've learned to recognize and interpret identities, a crucial skill in advanced mathematics. The ability to solve equations is a cornerstone of mathematical proficiency. It's a skill that transcends the classroom and finds applications in various fields, from science and engineering to finance and economics. By mastering the techniques discussed in this article, you've equipped yourself with a valuable tool for problem-solving and critical thinking. So, embrace the challenge of solving equations, and let the journey of mathematical discovery continue.