Solving The Linear Equation: 8/6 - (1/6)z + 6/3 = (2/6)z

by Andrew McMorgan 57 views

Hey math enthusiasts! Today, we're diving into solving a linear equation that might look a bit intimidating at first glance. But don't worry, we'll break it down step-by-step so it becomes super clear. Our equation is: 8/6 - (1/6)z + 6/3 = (2/6)z. Let's get started and tackle this equation together! We'll cover each step in detail, making sure you understand the logic behind every move. By the end of this article, you'll be a pro at solving similar equations. We will start by simplifying fractions and combining like terms. This is a fundamental step in solving any algebraic equation, and it helps to make the problem more manageable. Next, we will isolate the variable terms on one side of the equation and the constant terms on the other side. This is achieved by performing the same operations on both sides of the equation, ensuring that the equation remains balanced. Finally, we will solve for the variable by dividing both sides of the equation by the coefficient of the variable. This will give us the value of the variable that satisfies the equation. We'll also look at common mistakes to avoid and how to check your answers, so you can be confident in your solutions. So, grab your pencils and notebooks, and let's dive into the world of algebra!

1. Simplifying the Equation

First things first, let's simplify the equation to make it easier to work with. Our equation is 8/6 - (1/6)z + 6/3 = (2/6)z. To simplify, we'll focus on reducing the fractions and combining any constant terms. This involves several key steps, each designed to make the equation more manageable and easier to solve. The initial equation might seem daunting, but by breaking it down into smaller, more digestible parts, we can tackle it effectively. Remember, the goal is to transform the equation into a simpler form without changing its inherent mathematical meaning. This will pave the way for isolating the variable and finding its value. We'll start by simplifying the numerical fractions, then address the terms involving the variable 'z'. This methodical approach is crucial in solving any algebraic equation, especially when dealing with fractions and multiple terms. By following these steps, you'll be able to confidently solve this equation and similar ones in the future. Let’s dive in and see how it’s done!

1.1 Reducing Fractions

Okay, let's start by reducing the fractions. We have 8/6 and 6/3. Can we make these any simpler? Absolutely! When simplifying fractions, the goal is to reduce them to their lowest terms. This makes the numbers smaller and easier to work with, which in turn simplifies the overall equation. The process involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD. For instance, in the fraction 8/6, both 8 and 6 are divisible by 2. Similarly, in the fraction 6/3, both numbers are divisible by 3. Simplifying fractions is not just about making the numbers smaller; it's about making the equation clearer and more manageable. It's a fundamental step that can significantly reduce the complexity of the problem, making it easier to see the next steps. By starting with simplification, we set ourselves up for success in solving the equation efficiently. Let's see how these fractions look in their simplest forms!

  • 8/6 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 8/6 becomes 4/3.
  • 6/3 is a fraction that can be easily simplified as well. Both 6 and 3 are divisible by 3. Dividing both by 3, we get 2.

So, our equation now looks like this: 4/3 - (1/6)z + 2 = (2/6)z. See? Already a bit cleaner, right?

1.2 Combining Constant Terms

Next up, we need to combine the constant terms. These are the terms that don't have a 'z' attached to them. In our simplified equation, 4/3 - (1/6)z + 2 = (2/6)z, the constants are 4/3 and 2. Combining constants is a crucial step in simplifying the equation because it reduces the number of terms we need to deal with. This makes the equation less cluttered and easier to work with, ultimately leading to a smoother solving process. By adding or subtracting constants, we are essentially grouping like terms together, which is a fundamental principle in algebra. This step sets the stage for isolating the variable 'z' on one side of the equation. Let’s get these constants combined and make our equation even simpler!

To combine 4/3 and 2, we need to express 2 as a fraction with a denominator of 3. So, 2 is the same as 6/3. Now we can add them:

  • 4/3 + 6/3 = 10/3

Our equation is now: 10/3 - (1/6)z = (2/6)z. Looking much better, isn't it?

2. Isolating the Variable

Now, let's get to the heart of the matter: isolating the variable 'z'. This means we want all the terms with 'z' on one side of the equation and all the constants on the other. This is a crucial step in solving any algebraic equation because it separates the unknowns from the knowns, allowing us to determine the value of the variable. Isolating the variable involves using inverse operations to move terms across the equals sign. Whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the balance of the equation. This principle of maintaining balance is fundamental to algebraic manipulation. In our case, we'll need to move the term -(1/6)z to the right side of the equation and keep the constant term 10/3 on the left side. This will set us up to solve for 'z' in the next step. Let’s see how we can do this!

To do this, we'll add (1/6)z to both sides of the equation. Remember, whatever we do to one side, we do to the other to keep things balanced!

  • 10/3 - (1/6)z + (1/6)z = (2/6)z + (1/6)z
  • This simplifies to 10/3 = (2/6)z + (1/6)z

2.1 Combining 'z' Terms

Okay, time to combine the 'z' terms on the right side of the equation. We have (2/6)z + (1/6)z. Combining like terms is a fundamental step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power. In this case, both terms have the variable 'z' and the same denominator, which makes the addition straightforward. Combining the 'z' terms not only simplifies the equation but also brings us closer to isolating the variable and finding its value. This step is a crucial bridge between having multiple terms with 'z' and having a single term that we can easily solve for. By performing this addition, we reduce the complexity of the equation and make it easier to see the next steps. Let’s see how this addition works!

Since the fractions have the same denominator, we can simply add the numerators:

  • (2/6)z + (1/6)z = (2+1)/6 z = (3/6)z

So, our equation is now 10/3 = (3/6)z. We're getting closer!

2.2 Simplifying the 'z' Coefficient

Before we solve for 'z', let's simplify the coefficient (the number in front of 'z'). We have (3/6)z. Simplifying the coefficient of the variable is an important step because it reduces the fraction to its lowest terms, making the subsequent calculations easier and more straightforward. This simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD. In the case of 3/6, the GCD is 3, so dividing both numbers by 3 will simplify the fraction. By simplifying the coefficient, we make the equation cleaner and easier to interpret. This step sets us up for the final isolation of the variable, where we can determine its value by performing a single operation. Let’s simplify this coefficient and see how it looks!

3/6 can be simplified by dividing both the numerator and the denominator by 3:

  • 3/6 = 1/2

Now our equation is 10/3 = (1/2)z. Looking good!

3. Solving for 'z'

Alright, we've reached the final stage: solving for 'z'! This is where we isolate 'z' completely to find its value. We have the equation 10/3 = (1/2)z. To solve for 'z', we need to get rid of the fraction 1/2 that's multiplying 'z'. The most common way to do this is by multiplying both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. In our case, the reciprocal of 1/2 is 2/1, which is simply 2. By multiplying both sides of the equation by 2, we will effectively isolate 'z' on the right side, giving us the solution. This method is a fundamental technique in algebra and is widely used to solve equations involving fractions. Let’s go ahead and see how it’s done!

To get 'z' by itself, we need to multiply both sides of the equation by the reciprocal of 1/2, which is 2:

  • 2 * (10/3) = 2 * (1/2)z
  • This simplifies to 20/3 = z

So, z = 20/3. We did it!

3.1 Expressing the Solution

Now that we've found z = 20/3, let's express this solution in a way that's easy to understand. While 20/3 is a perfectly valid answer, it's often helpful to express it as a mixed number. A mixed number combines a whole number and a fraction, making it easier to visualize the value of the number. Converting an improper fraction (where the numerator is greater than the denominator) to a mixed number involves dividing the numerator by the denominator and expressing the remainder as a fraction. This transformation doesn't change the value of the number but presents it in a more accessible format. It's a common practice in mathematics to present solutions in their simplest and most understandable form, and expressing 20/3 as a mixed number is a great way to do that. Let’s see how we can convert this fraction!

20/3 is an improper fraction (the numerator is larger than the denominator). We can convert it to a mixed number:

  • 20 divided by 3 is 6 with a remainder of 2.
  • So, 20/3 = 6 2/3

Therefore, our final answer is z = 6 2/3. Nicely done!

4. Checking Your Answer

Okay, before we celebrate too much, let's check our answer to make sure it's correct. This is a crucial step in solving any equation. Checking your solution helps you catch any mistakes you might have made along the way. It gives you confidence in your answer and reinforces your understanding of the problem-solving process. The method we'll use is straightforward: we'll substitute our solution (z = 6 2/3) back into the original equation and see if both sides of the equation balance. If they do, we know we've solved the equation correctly. If they don't, it means we need to go back and review our steps to find any errors. This process of checking is not just a formality; it's a vital part of the problem-solving journey. Let’s dive in and make sure our solution is spot on!

To check, we'll substitute z = 20/3 back into the original equation: 8/6 - (1/6)z + 6/3 = (2/6)z

  • 8/6 - (1/6)(20/3) + 6/3 = (2/6)(20/3)
  • 8/6 - 20/18 + 6/3 = 40/18

Now, let's simplify and find a common denominator to add and subtract the fractions. The least common multiple of 6, 18, and 3 is 18.

  • (8/6)(3/3) - 20/18 + (6/3)(6/6) = 40/18
  • 24/18 - 20/18 + 36/18 = 40/18
  • (24 - 20 + 36)/18 = 40/18
  • 40/18 = 40/18

Yep, it checks out! Both sides of the equation are equal, so our solution z = 20/3 or 6 2/3 is correct.

Conclusion

Awesome job, you guys! We've successfully solved the equation 8/6 - (1/6)z + 6/3 = (2/6)z. We walked through each step, from simplifying fractions to isolating the variable and checking our answer. You've seen how breaking down a complex problem into smaller, manageable steps can make it much easier to solve. Remember, practice makes perfect, so keep tackling those equations and building your algebra skills. You've got this! The key takeaways from this problem are the importance of simplifying fractions, combining like terms, and isolating the variable. These techniques are fundamental in algebra and will help you solve a wide range of equations. Always remember to check your answers to ensure accuracy and build your confidence in your problem-solving abilities. We hope this article has been helpful and has boosted your understanding of solving linear equations. Keep up the great work, and you'll be an algebra whiz in no time!