Solving The Equation: 421.175(y*2) - 172.805(y-2) = 0

Hey guys! Today, we're diving into a fun math problem that some of you might find a bit tricky. We've got this equation: 421.175 * (y * 2) - 172.805 * (y - 2) = 0, and our mission, should we choose to accept it, is to find the value of 'y' that makes this equation true. Don't worry; we'll break it down step by step so it’s super easy to follow. So, grab your pencils, and let's get started!

Understanding the Equation

First off, let's take a good look at what we're dealing with. This equation involves decimals, multiplication, subtraction, and a variable 'y'. The goal here is to isolate 'y' on one side of the equation. This means we want to manipulate the equation, using mathematical operations, until we get something like y = some number. To achieve this, we'll need to simplify the equation by distributing, combining like terms, and then isolating 'y'. Remember, the golden rule of algebra is: whatever you do to one side of the equation, you must do to the other side to keep things balanced. It's like a mathematical seesaw – you gotta keep it level!

Before we even start crunching numbers, it's useful to have a strategy. Our plan of attack involves a few key steps:

1. Distribute: We'll get rid of those parentheses by multiplying the numbers outside the parentheses with each term inside.
2. Simplify: Next, we’ll simplify by performing any straightforward multiplication or combining similar terms.
3. Isolate 'y': Our main goal! We'll move all terms involving 'y' to one side of the equation and constants (plain numbers) to the other.
4. Solve for 'y': Finally, we’ll divide to get 'y' all by itself, giving us our solution.

With our battle plan ready, let’s roll up our sleeves and dive into the calculations!

Step-by-Step Solution

Okay, let's break this down into bite-sized pieces. Remember, the equation we’re tackling is:

421.175 * (y * 2) - 172.805 * (y - 2) = 0

1. Distribute

The first thing we need to do is get rid of those parentheses. This means we're going to distribute the numbers outside the parentheses to the terms inside. Let's take it one step at a time:

• 421.175 * (y * 2): Here, we can multiply 2 with 421.175 first, which gives us 842.35. So this part becomes 842.35 * y or 842.35y.
• -172.805 * (y - 2): We distribute -172.805 to both 'y' and '-2'.
  • -172.805 * y = -172.805y
  • -172.805 * -2 = 345.61 (Remember, a negative times a negative is a positive!).

So, after distributing, our equation looks like this:

842.35y - 172.805y + 345.61 = 0

2. Simplify

Now that we've distributed, let's simplify the equation by combining like terms. In this case, we have two terms with 'y' in them: 842.35y and -172.805y. We can combine these by simply adding (or subtracting, in this case) their coefficients:

842.35y - 172.805y = (842.35 - 172.805)y

Let’s do that subtraction: 842.35 - 172.805 = 669.545

So, our equation now looks like this:

669.545y + 345.61 = 0

3. Isolate 'y'

Alright, we're getting closer! Now we need to isolate 'y'. This means we want to get the term with 'y' on one side of the equation and everything else on the other side. To do this, we'll subtract 345.61 from both sides of the equation. This will cancel out the +345.61 on the left side:

669. 545y + 345.61 - 345.61 = 0 - 345.61

This simplifies to:

669.545y = -345.61

4. Solve for 'y'

Finally, the last step! To solve for 'y', we need to get 'y' all by itself. Right now, 'y' is being multiplied by 669.545. To undo this multiplication, we'll divide both sides of the equation by 669.545:

670. 545y / 669.545 = -345.61 / 669.545

This gives us:

y = -345.61 / 669.545

Now, let's do the division to get the numerical value of 'y':

y = -0.51617 (approximately)

So, we've found our solution! The value of 'y' that makes the original equation true is approximately -0.51617.

Verification

Before we call it a day, it's always a good idea to check our work. We can do this by plugging our solution back into the original equation and seeing if it holds true. Our original equation was:

421.175 * (y * 2) - 172.805 * (y - 2) = 0

Let's substitute y = -0.51617 into the equation:

421. 175 * (-0.51617 * 2) - 172.805 * (-0.51617 - 2) = 0

First, let's simplify inside the parentheses:

• -0.51617 * 2 = -1.03234
• -0.51617 - 2 = -2.51617

Now, substitute these values back into the equation:

422. 175 * (-1.03234) - 172.805 * (-2.51617) = 0

Perform the multiplications:

• 421.175 * -1.03234 = -434.799 (approximately)
• -172.805 * -2.51617 = 434.798 (approximately)

Now, let's add these results together:

-434. 799 + 434.798 = -0.001

This is very close to 0! The slight difference is due to rounding errors along the way. But overall, our solution checks out. We can confidently say that y ≈ -0.51617 is the solution to the equation.

Key Concepts Revisited

Let’s quickly recap the main concepts we used to solve this equation:

• Distribution: This is where we multiply a term outside parentheses with each term inside the parentheses. It’s crucial for getting rid of parentheses and simplifying the equation.
• Combining Like Terms: This involves adding or subtracting terms that have the same variable and exponent. It helps to make the equation cleaner and easier to work with.
• Isolating the Variable: This is the heart of solving equations. We move terms around so that the variable we want to find is on one side of the equation, and everything else is on the other side.
• Inverse Operations: We use inverse operations (like addition and subtraction, or multiplication and division) to undo operations and isolate the variable.
• Verification: Always a good practice! Plug your solution back into the original equation to make sure it works.

Common Mistakes to Avoid

Solving equations can be tricky, and it’s easy to make mistakes. Here are some common pitfalls to watch out for:

• Sign Errors: Be extra careful with negative signs. A small mistake with a sign can throw off the whole solution.
• Order of Operations: Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Do operations in the correct order.
• Distributing Negatives: When distributing a negative number, make sure to apply the negative sign to all terms inside the parentheses.
• Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine 3x and 5x, but you can’t combine 3x and 5x².
• Not Checking Your Work: Always, always, always check your solution by plugging it back into the original equation. It’s the best way to catch mistakes.

Practice Makes Perfect

So, there you have it! We’ve successfully solved a somewhat complex equation by breaking it down into manageable steps. Remember, the key to mastering algebra is practice. The more you practice, the more comfortable you'll become with these concepts, and the easier it will be to tackle even the toughest equations. So keep practicing, guys, and you'll be math whizzes in no time!

If you found this explanation helpful, give it a thumbs up, and don't forget to share it with your friends who might be struggling with equations too. And if you have any questions or want to see more examples, drop them in the comments below. Until next time, keep on solving!