Algebraic Equation: Solve For 's' In 4(s+6)=96

Hey guys, welcome back to Plastik Magazine! Today, we're diving headfirst into the fascinating world of algebra, tackling a problem that might look a little intimidating at first glance: Solve for $s$. $4(s+6)=96$. Don't sweat it, though! We're going to break this down step-by-step, making sure you understand every single move. Whether you're a math whiz or just looking to brush up on your skills, this guide is for you. We'll explore the fundamental principles of solving linear equations, the importance of order of operations, and how to isolate a variable. By the end of this, you'll be confidently solving similar equations, unlocking a whole new level of mathematical prowess. So, grab your notebooks, get comfy, and let's get our algebra on!

Understanding the Equation: What Are We Trying to Do?

Alright, let's get down to business. The equation we're working with is $4(s+6)=96$. Our main mission, should we choose to accept it, is to solve for $s$. What does that even mean? It means we need to find the specific value of $s$ that makes this entire statement true. Think of it like a puzzle where $s$ is the missing piece. We've got clues – the numbers and operations on either side of the equals sign – and we need to use them to reveal what $s$ represents. In algebra, the equals sign ($=$) is super important. It tells us that whatever is on the left side of it has the exact same value as whatever is on the right side. Our goal is to manipulate the equation, using established mathematical rules, until $s$ is all by itself on one side of the equals sign, and a number is on the other. This process is called isolating the variable. It's like gently nudging everything else away from $s$ until it's standing alone. This particular equation is a linear equation because the variable $s$ is raised to the power of 1. No exponents, no square roots of $s$, just $s$ itself. These are generally the first types of equations you'll encounter in algebra, and mastering them is key to tackling more complex problems down the line. We'll be using properties of equality, like the distributive property and inverse operations, to achieve our goal. The distributive property, for instance, helps us deal with numbers outside parentheses, and inverse operations (like addition and subtraction, or multiplication and division) are our trusty tools for unwinding the equation and getting to the heart of what $s$ is.

Step-by-Step Solution: Cracking the Code

Now for the fun part – actually solving it! Let's take it slow and steady. Our equation is $4(s+6)=96$. The first thing we notice is the number 4 sitting right outside the parentheses $(s+6)$. This means we need to multiply 4 by everything inside the parentheses. This is where the distributive property comes into play. We distribute the 4 to both $s$ and 6.

So, $4 * s$ becomes $4s$, and $4 * 6$ becomes $24$. Our equation now looks like this: $4s + 24 = 96$.

See? We've simplified it a bit already. Now, our mission is to get the term with $s$ (which is $4s$) all by itself. To do that, we need to get rid of that '+ 24'. What's the opposite, or inverse operation, of adding 24? You guessed it – subtracting 24! To keep our equation balanced, whatever we do to one side, we must do to the other side. So, we subtract 24 from both sides:

$(4s + 24) - 24 = 96 - 24$

On the left side, the $+24$ and $-24$ cancel each other out, leaving us with just $4s$. On the right side, $96 - 24$ equals $72$. So now our equation is:

$4s = 72$

We're so close, guys! $s$ is almost free. Right now, it's being multiplied by 4. What's the inverse operation of multiplying by 4? Dividing by 4! Again, we apply this to both sides of the equation to keep things fair:

$4s / 4 = 72 / 4$

On the left, the 4s cancel out, leaving $s$ all by its lonesome. On the right, $72$ divided by $4$ is $18$. And there we have it!

$s = 18$

Boom! We've successfully solved for $s$. It's like unlocking a secret code. Each step we took was guided by the principle of maintaining equality – keeping both sides of the equation the same. We used distribution to simplify, subtraction to isolate the variable term, and division to isolate the variable itself. It’s a systematic process, and once you get the hang of it, you’ll find yourself breezing through these kinds of problems.

Verification: Is Our Answer Correct?

So, we found that $s=18$. But how do we know for sure that this is the right answer? In math, we have a super handy technique called verification or checking your work. It's basically plugging our answer back into the original equation to see if it holds true. If it does, then we know we've nailed it! Let's try it with our equation: $4(s+6)=96$.

We're going to replace every $s$ in the equation with our solution, which is 18. So, it becomes:

$4(18+6) = 96$

Now, we follow the order of operations (PEMDAS/BODMAS – Parentheses/Brackets first!). Inside the parentheses, we have $18+6$. That equals $24$.

So now our equation looks like this:

$4(24) = 96$

Next, we perform the multiplication: $4 * 24$. If you do that calculation, you'll find that $4 * 24 = 96$.

So, the equation becomes:

$96 = 96$

And guess what? That statement is absolutely true! $96$ does indeed equal $96$. This means our value for $s$, which is $18$, is correct. This verification step is crucial, especially when you're dealing with more complex problems or when you're taking tests. It's your safety net, ensuring accuracy and boosting your confidence. Never skip this part if you want to be absolutely sure about your answers. It's a small step that can make a huge difference in your overall success with mathematics. It reinforces the understanding that an equation is a statement of balance, and the solution is the value that maintains that balance.

Why This Matters: Algebra in the Real World

Okay, so we've solved for $s$ and verified our answer. But you might be thinking, "Why should I care about solving equations like $4(s+6)=96$?" That's a totally fair question, guys. Algebra might seem like just a bunch of abstract symbols and rules, but it's actually a powerful tool that's used everywhere, all the time, often without us even realizing it. Think about it: whenever you need to figure something out, plan something, or even just budget your money, you're essentially doing algebra. For instance, if you're trying to figure out how much pizza you can afford for a party. Let's say you have a budget of $B$ dollars, and each pizza costs $P$ dollars. If you want to buy $N$ pizzas, you're dealing with an equation like $N * P = B$. You might need to solve for $N$ (how many pizzas) or $P$ (how much each pizza costs), or even $B$ (how much total you can spend). Or consider planning a road trip. You know the distance, and you have an idea of your average speed. You can use algebra to estimate how long the trip will take. The formula for distance is distance = speed × time ($d = s * t$). If you know $d$ and $s$, you can easily solve for $t$ to find your travel time. In science and engineering, algebra is fundamental. From calculating the trajectory of a rocket to designing a bridge, mathematical equations are the backbone of these professions. Even in everyday tasks like cooking, you might double a recipe, which involves multiplying all the ingredient amounts by 2 – a simple form of algebraic scaling. Understanding how to manipulate equations, isolate variables, and think logically about relationships between numbers is what algebra teaches you. It hones your problem-solving skills and teaches you to approach challenges in a structured, analytical way. So, while $4(s+6)=96$ might just be a problem on a page, the skills you use to solve it are applicable to a vast array of real-world scenarios, making you a more capable and confident problem-solver in all aspects of your life. It's about developing a mathematical mindset that can tackle complexity with clarity and precision.

Beyond the Basics: Other Ways to Solve

While the method we used – first distributing the 4 – is super common and effective, there's actually another way to tackle this equation, $4(s+6)=96$. It's all about choosing the path that feels most comfortable to you! Instead of distributing first, we can start by undoing the multiplication by 4. Remember, our goal is to isolate $s$. Right now, the entire expression $(s+6)$ is being multiplied by 4. To undo that multiplication, we can divide both sides of the equation by 4 right from the get-go.

So, starting with $4(s+6)=96$, we divide both sides by 4:

$[4(s+6)] / 4 = 96 / 4$

On the left side, the 4 divided by 4 cancels out, leaving us with just the expression inside the parentheses:

$s+6$

On the right side, $96$ divided by $4$ is $24$. So now our equation simplifies to:

$s+6 = 24$

See how this looks simpler? Now, we just need to get $s$ by itself. It's being added by 6. What's the inverse operation of adding 6? You know it – subtracting 6! We subtract 6 from both sides:

$(s+6) - 6 = 24 - 6$

On the left, the $+6$ and $-6$ cancel out, leaving $s$ alone.

On the right, $24 - 6 = 18$.

So, we get:

$s = 18$

And voilà! We arrive at the exact same answer, $s=18$. This alternative method highlights that there often isn't just one