Algebraic Equation: Solve For X

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of algebra, specifically tackling a linear equation that might look a bit intimidating at first glance: $-3x+3+2x=9x+3$. Don't sweat it, though! We're going to break this down step-by-step, making sure you understand every move we make. Our main goal here is to solve for x, which means we want to find the specific value of 'x' that makes this equation true. Think of 'x' as a mystery number we need to uncover.

Understanding the Equation

Let's start by looking at the equation itself: $-3x+3+2x=9x+3$. On the left side, we have terms involving 'x' ($-3x$ and $+2x$) and a constant term ($+3$). On the right side, we have another term involving 'x' ($9x$) and another constant term ($+3$). The '=' sign in the middle tells us that whatever is on the left side must have the exact same value as whatever is on the right side.

Our strategy to solve for x involves isolating 'x' on one side of the equation. This means we'll perform a series of operations (addition, subtraction, multiplication, division) on both sides of the equation to move all the 'x' terms to one side and all the constant terms to the other. It's like tidying up a room – we want all the 'x' stuff in one corner and all the number stuff in another.

Step 1: Simplify Both Sides

Before we start moving things around, it's always a good idea to simplify each side of the equation as much as possible. This means combining like terms. On the left side of our equation, we have $-3x$ and $+2x$. These are like terms because they both contain the variable 'x'. We can combine them by adding their coefficients: $-3 + 2 = -1$. So, $-3x + 2x$ simplifies to $-1x$, or just $-x$.

Our equation now looks like this: $-x + 3 = 9x + 3$.

Notice that the right side, $9x + 3$, is already simplified because $9x$ and $3$ are unlike terms (one has an 'x', the other doesn't), so they can't be combined further. Simplifying first helps make the subsequent steps much cleaner and reduces the chance of errors. It’s like prepping your ingredients before you start cooking – essential for a delicious result!

Step 2: Combine 'x' Terms

Now that both sides are simplified, our next mission is to get all the 'x' terms onto one side of the equation. You can choose to move them to the left or the right, but often it's easier to move them to the side where the coefficient of 'x' will end up being positive. In our case, we have $-x$ on the left and $9x$ on the right. If we add $x$ to both sides, we'll get $10x$ on the right, which is positive. This is generally preferred.

So, let's add $x$ to both sides of the equation:

$-x + 3 + x = 9x + 3 + x$

On the left side, $-x + x$ cancels out, leaving us with just $3$. On the right side, $9x + x$ combines to $10x$. So, the equation becomes:

$3 = 10x + 3$

See? We've successfully gathered all the 'x' terms on the right side. This is a huge step towards isolating our mystery variable!

Step 3: Isolate the 'x' Term

We're getting closer, guys! Our equation is now $3 = 10x + 3$. The 'x' term ($10x$) is on the right side, but it's still being added to $3$. To isolate the $10x$ term, we need to get rid of that $+3$. We do this by performing the opposite operation. Since $3$ is being added, we'll subtract $3$ from both sides of the equation to maintain balance.

$3 - 3 = 10x + 3 - 3$

On the left side, $3 - 3$ equals $0$. On the right side, $3 - 3$ cancels out, leaving us with just $10x$. So, the equation simplifies further to:

$0 = 10x$

We're almost there! The 'x' term is now isolated. This means $10x$ must equal $0$.

Step 4: Solve for 'x'

The final step to solve for x is to get 'x' completely by itself. Currently, 'x' is being multiplied by $10$ ($10x$). To undo multiplication, we use the inverse operation: division. We need to divide both sides of the equation by $10$.

rac{0}{10} = rac{10x}{10}

On the left side, $0$ divided by any non-zero number is always $0$. On the right side, rac{10x}{10} simplifies to just $x$ (because the $10$s cancel out).

So, we have:

$0 = x$

And there you have it! The solution to our equation is $x = 0$. We've successfully uncovered the mystery number!

Verification: Checking Our Answer

It's always a good practice, especially when you're learning, to check your answer to make sure it's correct. We do this by plugging our solution, $x=0$, back into the original equation: $-3x+3+2x=9x+3$.

Let's substitute $0$ for every $x$:

$-3(0) + 3 + 2(0) = 9(0) + 3$

Now, let's simplify both sides:

Left side: $-3(0) = 0$, and $2(0) = 0$. So, $0 + 3 + 0 = 3$.

Right side: $9(0) = 0$. So, $0 + 3 = 3$.

Since the left side ($3$) equals the right side ($3$), our solution $x=0$ is absolutely correct! This verification step confirms that our calculations were spot on and we've truly mastered solving this linear equation.

Conclusion

So there you have it, folks! We took the equation $-3x+3+2x=9x+3$, simplified it, combined like terms, isolated the variable, and found that $x=0$. Remember, the key principles in solving algebraic equations are to keep both sides balanced by performing the same operation on each side, and to use inverse operations to isolate the variable. Keep practicing these steps, and you'll become an algebra whiz in no time! What other equations are you guys struggling with? Let us know in the comments below! Until next time, keep experimenting and keep learning!