How To Solve For Y: 6y + 2x = 48

Hey mathletes! Ever find yourself staring at an equation like $6y + 2x = 48$ and wondering, "Alright, how do I even begin to isolate that pesky 'y'?" Don't sweat it, guys! Today, we're diving deep into the awesome world of algebra to crack this one open. We'll break it down step-by-step, making sure you understand every move. Remember, the goal here is to get 'y' all by itself on one side of the equals sign. It's like giving 'y' its own personal space, away from all the other numbers and variables. Think of it as untangling a knot – sometimes it takes a little patience and the right moves to get things straight. This equation, $6y + 2x = 48$, is a linear equation, meaning when you graph it, you get a straight line. But before we get to graphing, we gotta figure out what 'y' is worth in terms of 'x'. This skill is super important in math, and once you get the hang of it, you'll be solving all sorts of equations like a pro. So, grab your favorite thinking cap, maybe a calculator if you're feeling fancy, and let's get this equation solved! We're not just going to give you the answer; we're going to show you the why behind each step, so you can tackle similar problems with confidence. It’s all about building that problem-solving muscle, you know? We’ll keep it super clear and easy to follow, so no one gets left behind. Ready? Let's go!

Step 1: Identify Your Goal - Isolate 'y'

Alright, first things first, let's re-state our mission. We have the equation $6y + 2x = 48$, and our primary objective, our ultimate quest, is to get 'y' completely alone. That means we need to move everything that's currently hanging out with 'y' – in this case, the '6' that's multiplying it and the '+ 2x' that's being added to it – over to the other side of the equation. Think of the equals sign as a balanced scale. Whatever you do to one side, you must do to the other side to keep that scale perfectly balanced. If you add weight to one side, you have to add the same weight to the other. If you take something away, you take the same amount away from both. This principle is the golden rule of algebra, and it's what makes solving equations possible. For our equation, $6y + 2x = 48$, we see 'y' is being multiplied by 6, and then 2x is being added to that. To undo these operations, we'll use inverse operations. Addition is undone by subtraction, and multiplication is undone by division. We'll tackle these one by one, usually starting with the term that's being added or subtracted, before dealing with multiplication or division. This systematic approach ensures we don't mess up the balance of the equation. So, keep that goal in mind: get 'y' by itself. Every move we make will be a step towards that final, solitary 'y'. It’s like a puzzle, and we’re putting the pieces in the right place. We’re aiming for a final form that looks something like $y = [ ext{some expression involving x and numbers}]$. That's the dream, folks!

Step 2: Eliminate the '+ 2x' Term

Okay, team, let's get to work on our equation: $6y + 2x = 48$. Right now, '2x' is chilling on the same side as '6y'. To start isolating '6y', we need to get rid of that '+ 2x'. How do we do that? With its opposite operation, of course! The opposite of adding '2x' is subtracting '2x'. So, we're going to subtract '2x' from both sides of the equation. This is super important because, remember, we have to keep that scale balanced! If we only subtracted '2x' from the left side, the equation would be all wonky. So, here's what it looks like:

$6y + 2x - 2x = 48 - 2x$

On the left side, the '+ 2x' and the '- 2x' cancel each other out, leaving us with just '6y'. This is exactly what we wanted! On the right side, we now have '48 - 2x'. It's important to note that '48' and '-2x' are not 'like terms' (one is a constant, the other has a variable), so we can't combine them into a single number. We just leave them as they are for now. So, after this step, our equation has transformed into:

$6y = 48 - 2x$

See? We're one step closer to having 'y' all by itself. We've successfully removed the term that was being added to our 'y' term. This is a classic move in algebra – using inverse operations to simplify and isolate. It's like clearing the path so we can get to our main objective. Don't underestimate the power of this step; it sets up the next move perfectly. We’ve simplified the left side and adjusted the right side accordingly, maintaining the equality. This is the foundation for the next part of our solution. Keep your eyes on the prize: that lonely 'y'!

Step 3: Isolate 'y' by Dividing

Alright, we've made some serious progress! Our equation is now $6y = 48 - 2x$. We're so close, guys! We have '6y' on one side, and we want just 'y'. Currently, 'y' is being multiplied by 6. What's the opposite of multiplying by 6? You guessed it – dividing by 6! So, just like before, to keep our equation balanced, we need to divide every single term on both sides of the equation by 6. This is crucial: you must divide each term on the right side by 6, not just the '48' or the '-2x' individually.

Here’s how it looks:

$$\frac{6y}{6} = \frac{48 - 2x}{6}$$

Let's tackle the left side first. $\frac{6y}{6}$ simplifies beautifully to just 'y'. Hallelujah! We've done it! The 'y' is now by itself on the left side. Now, let's look at the right side: $\frac{48 - 2x}{6}$. We need to divide both '48' and '-2x' by 6:

$\frac{48}{6} - \frac{2x}{6}$

Let's simplify each of these fractions:

• $\frac{48}{6} = 8$
• $\frac{2x}{6}$ can be simplified by dividing both the numerator and the denominator by their greatest common factor, which is 2. So, $\frac{2x}{6} = \frac{1x}{3}$, or simply $\frac{x}{3}$.

Putting it all together, the right side becomes $8 - \frac{x}{3}$.

So, our final solved equation is:

$$y = 8 - \frac{x}{3}$$

And there you have it! We've successfully solved for 'y'. This is the solution to the equation in terms of 'x'. This means that for any value of 'x' you choose, you can plug it into $8 - \frac{x}{3}$ to find the corresponding 'y' value that satisfies the original equation $6y + 2x = 48$. Isn't that neat? We took a seemingly complex equation and, with a few logical steps, turned it into a clear expression for 'y'. This is the power of algebraic manipulation, and it's a skill that will serve you incredibly well as you continue your math journey. Practice this a few more times with different equations, and you'll be a whiz in no time!

Understanding the Solution

So, we ended up with $y = 8 - \frac{x}{3}$. What does this actually mean, guys? This equation is essentially a rule that tells us the relationship between 'x' and 'y' in the original equation, $6y + 2x = 48$. It tells us that for any pair of (x, y) values that make the original equation true, the 'y' value will always be equal to 8 minus one-third of the 'x' value. It's like a secret code between 'x' and 'y'! For example, let's pick an 'x' value. Say, $x = 3$. If we plug this into our solution:

$y = 8 - \frac{3}{3}$ $y = 8 - 1$ $y = 7$

So, the point (3, 7) should be a solution to the original equation. Let's check:

$6y + 2x = 48$ $6(7) + 2(3) = 48$ $42 + 6 = 48$ $48 = 48$

Boom! It works! See how our solution $y = 8 - \frac{x}{3}$ correctly predicts the 'y' value. Let's try another one. What if $x = 0$?

$y = 8 - \frac{0}{3}$ $y = 8 - 0$ $y = 8$

So, the point (0, 8) should also work. Check:

$6y + 2x = 48$ $6(8) + 2(0) = 48$ $48 + 0 = 48$ $48 = 48$

Nailed it again! This form of the equation, $y = 8 - \frac{x}{3}$, is often called the slope-intercept form when we're talking about linear equations, although it's written slightly differently here ($y = mx + b$). In our case, it's $y = -\frac{1}{3}x + 8$. The '8' is the y-intercept (where the line crosses the y-axis, which we found when $x=0$), and the '-1/3' is the slope (how steep the line is). Understanding this relationship is key to graphing lines and analyzing data. So, solving for 'y' isn't just about getting an answer; it's about revealing the underlying structure and relationships within the equation. It unlocks the ability to predict, check, and visualize. Pretty powerful stuff for one little equation, right? Keep practicing these steps, and you'll master this algebra game in no time!