Solving 6y - 20 = 2y - 4 A Step By Step Guide
Hey guys! Let's dive into solving a simple algebraic equation. It might seem tricky at first, but trust me, with a few basic steps, you'll be a pro in no time! Today, we're going to break down how to solve the equation 6y - 20 = 2y - 4. We'll go through each step meticulously, so you understand not just the how, but also the why behind every move. So, grab your pencils and let's get started!
Understanding the Basics of Algebraic Equations
Before we jump into the specifics of our equation, let's quickly recap what an algebraic equation actually is. Think of it as a balanced scale. On one side, we have an expression (like 6y - 20), and on the other side, we have another expression (like 2y - 4). The equals sign (=) tells us that both sides have the same value, keeping the scale balanced. Our goal is to find the value of the variable, in this case, 'y', that keeps this balance intact. To do this, we use inverse operations, which are just operations that undo each other – like addition and subtraction, or multiplication and division.
In algebraic equations, the goal is to isolate the variable on one side of the equation. This means we want to get 'y' all by itself on either the left or the right side. We achieve this by performing the same operations on both sides of the equation. This ensures that the equation remains balanced. For instance, if we subtract a number from one side, we must subtract the same number from the other side. This principle is crucial in solving equations accurately. Understanding this concept is key to mastering algebra. The beauty of algebra lies in its systematic approach. Every problem, no matter how complex it looks, can be broken down into smaller, manageable steps. So, don’t be intimidated by long equations; just take it one step at a time, and you'll be able to solve them.
Step-by-Step Solution of 6y - 20 = 2y - 4
Now, let's tackle our equation: 6y - 20 = 2y - 4. Remember our goal: to isolate 'y' on one side. Here's how we'll do it:
Step 1: Group the 'y' terms together
Our first mission is to gather all the terms containing 'y' on one side of the equation. Looking at 6y - 20 = 2y - 4, we see 'y' terms on both sides. A common strategy is to move the smaller 'y' term to the side with the larger 'y' term. In this case, we have 6y on the left and 2y on the right. Since 2y is smaller than 6y, we'll move it to the left side.
How do we move it? By using the inverse operation! Since 2y is being added (it's a positive term), we'll subtract 2y from both sides of the equation. This is crucial to maintain the balance. Our equation now looks like this:
6y - 20 - 2y = 2y - 4 - 2y
Simplifying both sides, we get:
4y - 20 = -4
See how we've grouped the 'y' terms? We're one step closer to isolating 'y'!
Step 2: Isolate the constant terms
Next, we need to deal with the constant terms, which are the numbers without any 'y' attached (in our case, -20 and -4). We want to get all the constant terms on the side opposite the 'y' term. Currently, we have -20 on the left side with the 4y term. To move it, we'll use the inverse operation. Since 20 is being subtracted, we'll add 20 to both sides of the equation:
4y - 20 + 20 = -4 + 20
Simplifying, we get:
4y = 16
Fantastic! We've successfully isolated the 'y' term on one side and the constant term on the other.
Step 3: Solve for 'y'
We're almost there! Now we have 4y = 16. This means 4 times 'y' equals 16. To find 'y', we need to undo the multiplication. The inverse operation of multiplication is division, so we'll divide both sides of the equation by 4:
(4y) / 4 = 16 / 4
Simplifying, we get:
y = 4
And there you have it! We've solved the equation. The value of 'y' that satisfies the equation 6y - 20 = 2y - 4 is 4.
Verifying the Solution
It's always a good idea to check our answer to make sure it's correct. To do this, we substitute our solution, y = 4, back into the original equation:
6y - 20 = 2y - 4
Substitute y = 4:
6(4) - 20 = 2(4) - 4
Simplify:
24 - 20 = 8 - 4
4 = 4
The left side equals the right side! This confirms that our solution, y = 4, is indeed correct. This verification step is crucial in ensuring the accuracy of your solution. By plugging the value back into the original equation, you can confidently say that you have solved the problem correctly. It also helps in understanding the balance and equality that algebraic equations represent.
Conclusion: The Correct Answer
So, after walking through the steps together, we found that the solution to the equation 6y - 20 = 2y - 4 is y = 4. This corresponds to option B in the multiple-choice answers you provided. Remember, guys, practice makes perfect! The more equations you solve, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a part of learning. Just keep practicing, and you'll master algebra in no time.
Algebraic equations might seem intimidating at first, but with a clear understanding of the basic principles and a systematic approach, they become much more manageable. Remember to focus on isolating the variable by using inverse operations and always verify your solution to ensure accuracy. Keep up the great work, and you'll become a pro at solving equations! And most importantly, don't forget to have fun while learning!
Practice Problems
To further solidify your understanding, here are a few practice problems you can try:
- 3x + 5 = x - 1
- 8z - 12 = 4z + 8
- 5a + 7 = 2a - 2
Try solving these equations using the same steps we discussed. Check your answers by substituting your solutions back into the original equations. Happy solving, guys!
