You know, sometimes the simplest-looking equations can feel like a bit of a puzzle, can't they? Take this one: 6y - 9 = 3y + 7. It looks straightforward, but if you're not used to wrestling with algebra, it might make you pause. I remember when I first started dabbling in math beyond basic arithmetic; variables like 'y' felt like secret codes. But the beauty of algebra is that it provides a systematic way to crack those codes.
Let's break down this particular equation, shall we? The goal here is to isolate that elusive 'y' and figure out what number it represents. Think of it like a balancing act. Whatever you do to one side of the equals sign, you absolutely must do to the other to keep things fair and true.
Our starting point is: 6y - 9 = 3y + 7.
First off, we want to gather all the terms with 'y' on one side and all the plain numbers (constants) on the other. It's generally easier to move the smaller 'y' term. So, let's subtract '3y' from both sides. Why? Because 6y minus 3y leaves us with 3y, and we're trying to simplify.
So, 6y - 3y - 9 = 3y - 3y + 7. This tidies up to: 3y - 9 = 7.
Now, we've got the 'y' terms on the left. Let's get that '-9' out of the way. To do that, we add 9 to both sides. This is the inverse operation of subtracting 9, and it cancels out the -9 on the left.
Adding 9 to both sides gives us: 3y - 9 + 9 = 7 + 9. And that simplifies to: 3y = 16.
We're almost there! We have '3y', which means 3 times 'y'. To find out what just 'y' is, we need to do the opposite of multiplying by 3, which is dividing by 3. Again, we do this to both sides.
Dividing both sides by 3, we get: 3y / 3 = 16 / 3. And there it is: y = 16/3.
It's always a good idea to check your work, right? Let's plug '16/3' back into the original equation to see if it holds true.
On the left side: 6 * (16/3) - 9. 6 divided by 3 is 2, so that's 2 * 16 - 9 = 32 - 9 = 23.
On the right side: 3 * (16/3) + 7. 3 divided by 3 is 1, so that's 1 * 16 + 7 = 16 + 7 = 23.
See? Both sides equal 23. The equation balances perfectly. So, y = 16/3 is indeed the correct solution. It’s a satisfying feeling when the numbers finally click into place, isn't it?
