Solve The Equation: -4(2x+3) = 2x+6-(8x+2)

Understanding the Challenge of Algebraic Equations

When we first encounter algebraic equations, especially those with parentheses and multiple terms on both sides, it can feel a bit like staring at a puzzle. The goal, however, is always the same: to find the value of the unknown variable (in this case, 'x') that makes the entire equation true. This solution to the equation is like the key that unlocks the mystery, ensuring that both sides of the equality are perfectly balanced. Many students find themselves asking, "What is the solution to the equation?" when faced with expressions like $-4(2x+3)=2x+6-(8x+2)$. It's a common question, and the answer lies in a systematic approach of simplification and isolation. We need to carefully unwrap the parentheses, combine like terms, and gradually move all the 'x' terms to one side of the equation and all the constant numbers to the other. This process, while sometimes tedious, is fundamental to mastering algebra and building a strong foundation for more complex mathematical concepts. The beauty of mathematics is that there's a logical flow to these operations, and once you understand the rules, solving these equations becomes less daunting and more like a rewarding game of strategy. We'll break down this specific equation step-by-step, demystifying the process and highlighting the key principles that make it all work.

Step-by-Step Solution: Simplifying the Equation

Let's dive into the nitty-gritty of finding the solution to the equation $-4(2x+3)=2x+6-(8x+2)$. Our first move is to tackle the parentheses. On the left side, we have $-4$ multiplying the entire expression $(2x+3)$. This means we need to distribute the $-4$ to both $2x$ and $3$:

$-4 \times 2x = -8x$ $-4 \times 3 = -12$

So, the left side of our equation becomes $-8x - 12$. Now, let's turn our attention to the right side: $2x+6-(8x+2)$. Here, we have a minus sign in front of the parentheses $(8x+2)$. This minus sign acts like a $-1$ that needs to be distributed to both $8x$ and $2$.

$-(8x) = -8x$ $-(2) = -2$

So, the right side simplifies to $2x + 6 - 8x - 2$.

Now, let's combine the like terms on the right side. We have $2x$ and $-8x$, which combine to give us $(2-8)x = -6x$. We also have the constants $6$ and $-2$, which combine to give us $6 - 2 = 4$.

Therefore, the right side of our equation simplifies to $-6x + 4$.

Putting it all together, our simplified equation now looks like this:

$-8x - 12 = -6x + 4$

This is a significant step because we've eliminated the parentheses and combined like terms, making the equation much easier to manage. This simplification process is crucial. It's like clearing the clutter before you can see the actual problem. Each step taken is a deliberate move towards isolating the variable, and understanding why we perform these operations – like distributing and combining like terms – is key to building confidence in solving any algebraic equation. Remember, the goal is to maintain the balance of the equation at every stage. Whatever operation you perform on one side, you must perform on the other to keep the equality true.

Isolating the Variable: Finding the Value of x

With our simplified equation $-8x - 12 = -6x + 4$, we're now ready to isolate the variable 'x'. The objective is to get all the 'x' terms on one side of the equation and all the constant terms on the other. Let's start by moving the 'x' terms. We can add $8x$ to both sides of the equation. This will cancel out the $-8x$ on the left side:

$-8x - 12 + 8x = -6x + 4 + 8x$

This simplifies to:

$-12 = (-6x + 8x) + 4$ $-12 = 2x + 4$

Now, we need to move the constant term, which is $4$, from the right side to the left side. We can do this by subtracting $4$ from both sides of the equation:

$-12 - 4 = 2x + 4 - 4$

This simplifies to:

$-16 = 2x$

We're almost there! The variable 'x' is currently being multiplied by $2$. To get 'x' by itself, we need to perform the inverse operation, which is division. We will divide both sides of the equation by $2$:

$-16 / 2 = 2x / 2$

This gives us our final answer:

$-8 = x$

So, the solution to the equation $-4(2x+3)=2x+6-(8x+2)$ is $x = -8$. This process of isolating the variable is fundamental in algebra. It relies on the principle of inverse operations: addition undoes subtraction, multiplication undoes division, and vice versa. By applying these operations strategically to both sides of the equation, we systematically peel away the layers around 'x' until it stands alone, revealing its true value. Each step taken – adding $8x$ to both sides, subtracting $4$ from both sides, and dividing by $2$ – is designed to maintain the equality while inching closer to the solution.

Verification: Checking Your Solution

It's always a good practice to verify your solution to an algebraic equation. This means plugging the value you found for 'x' back into the original equation to ensure that both sides are equal. Our solution is $x = -8$. Let's substitute $-8$ for 'x' in the original equation: $-4(2x+3)=2x+6-(8x+2)$.

Left side: $-4(2(-8)+3)$ $-4(-16+3)$ $-4(-13)$ $52$

Right side: $2(-8)+6-(8(-8)+2)$ $-16+6-(-64+2)$ $-16+6-(-62)$ $-16+6+62$ $-10+62$ $52$

Since the left side ($52$) equals the right side ($52$), our solution $x = -8$ is correct! This verification step is incredibly important. It not only confirms your answer but also reinforces your understanding of how equations work. If the numbers on both sides don't match, it indicates a mistake was made somewhere in the simplification or isolation process, prompting you to go back and review your steps. This self-checking mechanism is a powerful tool for learning and ensuring accuracy in mathematical problem-solving. It transforms the process from a blind guess to a methodical confirmation of the derived result. For more complex problems, this verification can save a lot of time and frustration, solidifying your grasp on the underlying algebraic principles. It's a testament to the logical structure of mathematics that a single, correct value for 'x' can bring such perfect balance to a seemingly intricate equation.

Conclusion: Mastering Algebraic Equations

We've successfully navigated the process of finding the solution to the equation $-4(2x+3)=2x+6-(8x+2)$, arriving at $x = -8$. This journey involved several key algebraic techniques: distribution to remove parentheses, combining like terms to simplify expressions, and the strategic use of inverse operations to isolate the variable. Each step builds upon the last, demonstrating the logical and sequential nature of algebra. The ability to solve such equations is not just about getting the right answer; it's about developing critical thinking skills, logical reasoning, and a systematic approach to problem-solving that can be applied to countless situations beyond mathematics. Remember, the core principle is to maintain balance – whatever you do to one side of the equation, you must do to the other. This ensures that the equality holds true throughout the entire process. Practice is key to building confidence and fluency in algebra. The more equations you solve, the more intuitive these steps will become. Don't be discouraged by mistakes; they are often valuable learning opportunities that highlight areas needing more attention. With consistent effort and a clear understanding of the fundamental rules, you'll find yourself tackling more complex algebraic challenges with ease and confidence. The world of mathematics is vast and rewarding, and mastering basic algebraic equations is a significant step in exploring its depths.

For further exploration into the principles of solving linear equations and other algebraic concepts, you can visit trusted resources like Khan Academy or Wolfram MathWorld.