Solving The Equation: $12-(4y+8)=0.5(8y-16)$

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Hey guys! Today, we are diving into a fun math problem that involves solving a linear equation. Linear equations are super important in algebra, and mastering them will help you tackle more complex problems later on. So, let’s break down the equation $12-(4y+8)=0.5(8y-16)$ step by step. We'll make sure everyone understands each stage of the process, so grab your pencils and let's get started!

Understanding the Equation

First off, let's take a good look at our equation: $12-(4y+8)=0.5(8y-16)$. This equation might seem a little intimidating at first, but don't worry! We can handle it. The goal here is to find the value of 'y' that makes the equation true. To do this, we need to simplify the equation by performing some algebraic operations. Remember, whatever we do on one side of the equation, we must also do on the other side to keep everything balanced. Think of it like a see-saw – if you add weight to one side, you need to add the same weight to the other side to keep it level.

Breaking Down the Terms

Before we jump into the actual solving, let's quickly identify the different parts of the equation. We have constants (like 12 and 8), variables (that's our 'y'), and coefficients (the numbers multiplied by the variable, such as 4 and 0.5). We also have parentheses, which tell us to perform the operations inside them first. Understanding these components is crucial because it helps us decide which steps to take in what order. It's like having a map before starting a journey; knowing the terrain makes the trip much smoother. We also need to remember the order of operations (PEMDAS/BODMAS), which tells us to handle Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Keeping PEMDAS/BODMAS in mind will ensure we tackle the equation in the correct sequence.

Step-by-Step Solution

Now, let’s get our hands dirty and solve the equation. We'll go through each step meticulously, so you can follow along easily.

Step 1: Distribute the Constants

The first thing we need to do is get rid of those parentheses. To do this, we'll distribute the constants outside the parentheses to the terms inside. On the left side of the equation, we have $12 - (4y + 8)$. Notice the minus sign in front of the parentheses? It's like multiplying the entire expression inside the parentheses by -1. So, we distribute -1 to both $4y$ and $8$.

• $12 - (4y + 8) = 12 - 4y - 8$

On the right side, we have $0.5(8y - 16)$. Here, we multiply 0.5 by both $8y$ and $-16$.

• $0.5(8y - 16) = 4y - 8$

So, after distributing, our equation looks like this: $12 - 4y - 8 = 4y - 8$.

Step 2: Combine Like Terms

Next, we want to simplify each side of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power (or are just constants). On the left side, we have the constants $12$ and $-8$, which we can combine:

• $12 - 8 = 4$

So, the left side simplifies to $4 - 4y$. The right side, $4y - 8$, already has no like terms to combine. Our equation now looks like this: $4 - 4y = 4y - 8$.

Step 3: Isolate the Variable Terms

Now, we want to get all the terms with 'y' on one side of the equation and all the constants on the other side. Let’s move the $-4y$ term from the left side to the right side. To do this, we add $4y$ to both sides of the equation:

• $4 - 4y + 4y = 4y - 8 + 4y$
• This simplifies to $4 = 8y - 8$

Step 4: Isolate the Constant Terms

Next, let's move the constant term $-8$ from the right side to the left side. We do this by adding $8$ to both sides:

• $4 + 8 = 8y - 8 + 8$
• This simplifies to $12 = 8y$

Step 5: Solve for 'y'

Finally, to solve for 'y', we need to get 'y' by itself. It's currently being multiplied by 8, so we'll do the opposite operation: divide both sides by 8:

• $12 / 8 = 8y / 8$
• This simplifies to $y = 12/8$

We can further simplify the fraction $12/8$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

• $y = (12 ÷ 4) / (8 ÷ 4)$
• So, $y = 3/2$ or $y = 1.5$

Final Answer

Therefore, the solution to the equation $12-(4y+8)=0.5(8y-16)$ is $y = 3/2$ or $1.5$. We did it! 🎉

Checking Our Solution

To make sure we've got the right answer, it's always a good idea to check our solution. We do this by substituting our value for 'y' back into the original equation and seeing if both sides are equal.

Original equation: $12 - (4y + 8) = 0.5(8y - 16)$

Substitute $y = 1.5$:

• $12 - (4(1.5) + 8) = 0.5(8(1.5) - 16)$
• $12 - (6 + 8) = 0.5(12 - 16)$
• $12 - 14 = 0.5(-4)$
• $-2 = -2$

Both sides are equal, so our solution $y = 1.5$ is correct. High five! 🖐️

Common Mistakes to Avoid

When solving equations, there are a few common pitfalls that students often fall into. Let's look at some of these so you can steer clear of them!

Forgetting to Distribute the Negative Sign

One of the most frequent errors is not properly distributing the negative sign when dealing with parentheses. Remember, when you have a minus sign in front of parentheses, it's like multiplying the entire expression inside by -1. Make sure each term inside the parentheses gets multiplied by -1. For example, in the expression $12 - (4y + 8)$, it's essential to distribute the negative sign to both $4y$ and $+8$, resulting in $12 - 4y - 8$. Overlooking this step can lead to incorrect solutions.

Incorrectly Combining Like Terms

Another common mistake is combining terms that aren't actually like terms. Remember, like terms have the same variable raised to the same power. You can't combine a term with 'y' and a constant term. For instance, in the expression $4 - 4y$, you cannot combine $4$ and $-4y$ because they are different types of terms. Always double-check that you're only adding or subtracting terms that share the same variable and exponent.

Not Following the Order of Operations

The order of operations (PEMDAS/BODMAS) is crucial in solving equations correctly. Make sure you handle parentheses first, then exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). Skipping or misinterpreting the order of operations can throw off your entire solution. For example, in the expression $0.5(8y - 16)$, you must first perform the distribution of 0.5 before you attempt any other operations.

Not Performing the Same Operation on Both Sides

The golden rule of equation solving is that whatever you do on one side of the equation, you must also do on the other side. This keeps the equation balanced. If you add a number to one side, you must add the same number to the other side. If you divide by a number on one side, you must divide by the same number on the other side. Failing to maintain this balance will result in an incorrect solution. Always be meticulous in applying operations to both sides of the equation.

Skipping the Check Step

Finally, a very common mistake is not checking your solution. Always substitute your answer back into the original equation to verify that it makes the equation true. This simple step can catch errors and give you confidence in your solution. If you find that your solution doesn't satisfy the original equation, you'll know to go back and check your work. Checking your solution is a crucial part of the problem-solving process.

Practice Makes Perfect

Solving equations is like riding a bike – the more you practice, the better you'll get. Try solving similar equations to reinforce your understanding. Don't be afraid to make mistakes; they're part of the learning process. And remember, if you get stuck, go back and review the steps we've discussed. Math is a journey, not a destination, and every problem you solve makes you a little bit stronger.

Additional Practice Problems

Here are a few more equations you can try solving on your own. Feel free to share your solutions in the comments!

1. $3(x + 2) = 15$
2. $2y - 5 = 3y + 7$
3. $4(2a - 1) = 2(a + 5)$

These practice problems will help you sharpen your skills and build confidence. Remember to follow the same step-by-step approach we used in this article, and don't forget to check your solutions!

Conclusion

So, there you have it! We've successfully solved the equation $12-(4y+8)=0.5(8y-16)$. We broke it down step by step, talked about common mistakes, and even gave you some extra practice problems. Remember, math might seem tough sometimes, but with a little effort and the right approach, you can conquer any equation. Keep practicing, stay curious, and never stop learning. Until next time, happy solving! 👍

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