Solving $12(5.5-1.75x)+2.4=4(4x+12)+5.6$ For X

Let's dive into solving this equation step by step! Equations like this might seem intimidating at first, but by breaking them down, we can find the value of $x$. This article provides a detailed walkthrough on how to solve the equation $12(5.5-1.75x)+2.4=4(4x+12)+5.6$ for $x$. We'll cover each step meticulously, ensuring you understand the process and can confidently tackle similar problems in the future. Understanding the order of operations (PEMDAS/BODMAS) is crucial. We'll start with distribution, then combine like terms, and isolate $x$ to find our solution. These algebraic manipulations are fundamental in mathematics, and mastering them will significantly enhance your problem-solving abilities. So, grab your pencil and paper, and let's get started!

Step-by-Step Solution

1. Distribute the constants

The first step in solving the equation is to distribute the constants outside the parentheses to the terms inside. This means multiplying 12 by both 5.5 and -1.75x on the left side of the equation, and multiplying 4 by both 4x and 12 on the right side. Distribution is a fundamental algebraic technique that allows us to simplify expressions and equations. By multiplying each term inside the parentheses by the constant outside, we eliminate the parentheses and make the equation easier to manipulate. This process ensures that we maintain the equality of the equation while rearranging terms. This initial distribution is crucial because it sets the stage for combining like terms and isolating the variable $x$. Let's see how this looks:

• Left side: $12 * 5.5 = 66$ and $12 * -1.75x = -21x$
• Right side: $4 * 4x = 16x$ and $4 * 12 = 48$

So, the equation becomes:

$66 - 21x + 2.4 = 16x + 48 + 5.6$

2. Combine like terms on each side

Now, we need to combine the like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power (or are constants). On the left side, we have the constants 66 and 2.4, which can be added together. On the right side, we have the constants 48 and 5.6, which can also be added. Combining like terms simplifies the equation further, making it easier to isolate the variable. This step reduces the number of terms we need to deal with, which in turn reduces the complexity of the equation. It's a crucial step in organizing the equation and moving closer to the solution. Let’s perform the addition:

• Left side: $66 + 2.4 = 68.4$
• Right side: $48 + 5.6 = 53.6$

The equation now looks like this:

$68.4 - 21x = 16x + 53.6$

3. Move variable terms to one side

The next step is to move all the terms containing $x$ to one side of the equation and all the constants to the other side. To do this, we can add $21x$ to both sides of the equation. This will eliminate the $-21x$ term on the left side and move the $x$ term to the right side. Moving variable terms to one side is a key step in isolating the variable and solving for its value. It allows us to group the $x$ terms together, making it possible to eventually have a single term with $x$ on one side of the equation. This rearrangement is a fundamental algebraic technique used in solving various types of equations.

Adding $21x$ to both sides gives us:

$68.4 - 21x + 21x = 16x + 21x + 53.6$

Which simplifies to:

$68.4 = 37x + 53.6$

4. Move constant terms to the other side

Now, we need to move the constant term 53.6 from the right side to the left side. We can do this by subtracting 53.6 from both sides of the equation. This will leave the $37x$ term isolated on the right side. Just as we moved variable terms, moving constant terms is essential for isolating the variable and finding its value. This process ensures that all constants are grouped together on one side of the equation, simplifying the remaining steps. It's a standard technique used in algebra to solve for unknowns.

Subtracting 53.6 from both sides gives us:

$68.4 - 53.6 = 37x + 53.6 - 53.6$

Which simplifies to:

$14.8 = 37x$

5. Isolate x

Finally, to isolate $x$, we need to divide both sides of the equation by the coefficient of $x$, which is 37. This will give us the value of $x$. Isolating $x$ is the ultimate goal of solving the equation. By performing the necessary operations (in this case, division), we get $x$ by itself on one side of the equation, revealing its value. This step represents the culmination of all the previous steps, leading to the solution of the equation.

Dividing both sides by 37 gives us:

$14.8 / 37 = 37x / 37$

Which simplifies to:

$x = 0.4$

Final Answer

Therefore, the solution to the equation $12(5.5-1.75x)+2.4=4(4x+12)+5.6$ is $x = 0.4$. This corresponds to option A.

Conclusion

Solving algebraic equations involves a systematic approach. By carefully following the steps of distribution, combining like terms, and isolating the variable, we can find the solution. Practice is key to mastering these skills, so keep working on similar problems! Remember, the more you practice, the more comfortable you'll become with these algebraic manipulations. Each equation is a puzzle, and the process of solving it hones your problem-solving skills. Keep up the great work, and you'll become a master of algebra in no time!

For further learning and practice on algebraic equations, you can visit Khan Academy's Algebra I section. They offer numerous exercises and explanations to help you strengthen your understanding.