Kenya's Equation: Find The Solution!

Hey guys, let's dive into a cool math problem that Kenya tackled! We're talking about solving algebraic equations, and this one's a doozy. The equation we're looking at is:

$-6(x-2)+3 x=-3(x+3)+21$

Our mission, should we choose to accept it, is to figure out what x is. Is it a specific number, does it have no answer, or is it every number? Let's break it down step-by-step, shall we? This isn't just about crunching numbers; it's about understanding the logic behind each move. Think of it like a puzzle where each piece needs to fit perfectly. We'll use our trusty distributive property and combine like terms, those algebraic buddies that hang out together. Ready to get your math on?

Unpacking the Equation: The First Steps

Alright, let's get serious about this equation: $-6(x-2)+3 x=-3(x+3)+21$. The first thing we gotta do is simplify both sides of the equation. This means we need to get rid of those pesky parentheses using the distributive property. Remember that? It's like when you multiply a number outside the parentheses by each term inside. So, on the left side, we've got $-6$ multiplied by $(x-2)$. That gives us $-6*x$ and $-6*(-2)$. Boom! That's $-6x$ and $+12$. Now, we still have that $+3x$ hanging around, so the left side becomes $-6x + 12 + 3x$.

Now, let's hop over to the right side. We've got $-3$ multiplied by $(x+3)$. That's $-3*x$ and $-3*3$. So, we get $-3x$ and $-9$. Don't forget the $+21$ that's just chilling there. So, the right side is $-3x - 9 + 21$.

So, after this first distribution, our equation looks like this:

$-6x + 12 + 3x = -3x - 9 + 21$

See? We're already making progress. It's all about tackling one part at a time. Don't let the look of the equation intimidate you. We're just applying basic rules. Think of it as peeling an onion; you take off one layer at a time to get to the core. And that core, my friends, is the solution we're looking for. This step is super crucial because it makes the equation much cleaner and easier to work with. Without this simplification, trying to isolate x would be a nightmare. We've successfully applied the distributive property, which is a fundamental skill in algebra. It's like having the right tools in your toolbox; you can't build anything without them. So, give yourself a pat on the back, you've just conquered the first hurdle in solving this equation!

Combining Like Terms: Making It Even Simpler

Okay, we've distributed, and now our equation is $-6x + 12 + 3x = -3x - 9 + 21$. The next logical step, guys, is to combine like terms on each side of the equation. What does that mean? It means we group together all the terms that have an x in them, and we group together all the constant numbers (the ones without an x).

Let's look at the left side: $-6x + 12 + 3x$. We have two x terms: $-6x$ and $+3x$. If we combine them, $-6x + 3x$, what do we get? That's right, $-3x$. We still have the $+12$. So, the simplified left side is -3x + 12.

Now, let's move to the right side: $-3x - 9 + 21$. We have one x term: $-3x$. And we have two constant terms: $-9$ and $+21$. If we combine these constants, $-9 + 21$, we get $+12$. So, the simplified right side is -3x + 12.

So, after combining like terms, our equation now looks super neat:

-3x + 12 = -3x + 12

Can you believe it? We've simplified both sides so much that they look identical! This is a really interesting point in solving equations. It tells us something special is going on. We've applied the distributive property and combined like terms, which are foundational algebraic techniques. These steps are essential for isolating the variable and finding its value. By grouping similar terms, we reduce the complexity of the equation, making it easier to see the relationships between the variables and constants. This process ensures that we're not missing any opportunities to simplify and that we're systematically moving towards a solution. The elegance of algebra lies in these structured steps, transforming complex expressions into manageable ones. This current form of the equation is a direct result of accurate application of these rules. Now, what does this identical form actually mean for our solution? Let's find out in the next section!

The Moment of Truth: What Does It Mean?

We've arrived at this point: -3x + 12 = -3x + 12. This is where things get really interesting, guys. We want to isolate x, right? So, let's try to get all the x terms on one side and the numbers on the other. We can do this by adding $3x$ to both sides of the equation.

If we add $3x$ to the left side, we get $(-3x + 12) + 3x$. The $-3x$ and $+3x$ cancel each other out, leaving us with just $12$.

If we add $3x$ to the right side, we get $(-3x + 12) + 3x$. Again, the $-3x$ and $+3x$ cancel each other out, leaving us with just $12$.

So, after adding $3x$ to both sides, our equation becomes:

12 = 12

Now, what on earth does 12 = 12 mean? This is a true statement, right? $12$ is always equal to $12$, no matter what value x has. This means that the original equation is true for any value of x you can think of. Whether x is $5$, $-10$, $100$, or even a fraction, the equation will always hold true.

This special situation is called infinitely many solutions. It's like saying the equation is true for all real numbers. We didn't get a specific value for x because the equation simplifies to a statement that is always true. This happens when both sides of the equation are identical after simplification, which is exactly what we found. The process of distributing and combining like terms is fundamental. When these steps lead to an identity like $12=12$, it signifies that the original equation is an identity itself. This means that any value substituted for the variable will satisfy the equation. It's a powerful concept in algebra, demonstrating that not all equations have a single, unique solution. Some, like this one, embrace all possibilities!

Final Answer and What It Means for Kenya's Problem

So, to recap, Kenya's equation $-6(x-2)+3 x=-3(x+3)+21$ simplified down to $12 = 12$. This is a true statement, and it means that the equation is true for infinitely many solutions.

Let's quickly recap the options Kenya was given:

A. $-4$ B. $12$ C. no solution D. infinitely many solutions

Based on our work, the correct answer is D. infinitely many solutions.

It's important to understand what this means. When an equation has infinitely many solutions, it's not that there's no answer, but rather that every possible number is an answer. This is a key concept in algebra, differentiating it from equations that have a single solution (like $2x = 4$, where $x=2$) or no solution (like $x+1 = x+2$, which simplifies to $1=2$, a false statement).

Kenya solved this equation by diligently applying the distributive property and then combining like terms. These fundamental algebraic techniques allowed her to simplify both sides of the equation. When both sides turned out to be identical ($-3x + 12 = -3x + 12$), it led to the conclusion of infinite solutions. This outcome highlights the diverse nature of algebraic equations and the importance of recognizing different types of solutions. So, next time you see an equation that simplifies to a true statement (like $5=5$ or $0=0$), you'll know exactly what it means: infinitely many solutions! Keep practicing these steps, guys, and you'll be solving equations like a pro in no time. Math is all about understanding these patterns and properties!