Solving For X: A Step-by-Step Guide To The Equation

Hey math enthusiasts! Ever get stuck staring at an equation, wondering where to even begin? Don't worry, we've all been there. Today, we're going to break down a specific equation and show you exactly how to solve for x. We'll tackle the equation $-4+9.4(-2 x+1)=2(-7 x+1)+9.4$ step-by-step, so you can follow along and learn the process. Think of this as your friendly guide to conquering algebraic challenges!

Understanding the Equation

Before we dive into the solution, let's take a moment to understand what the equation is telling us. In this equation, $-4+9.4(-2 x+1)=2(-7 x+1)+9.4$, we have a variable, x, which represents an unknown value. Our goal is to isolate x on one side of the equation to determine its value. The equation involves several mathematical operations, including multiplication, addition, and distribution. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order will be crucial as we simplify the equation.

Initial Assessment of the Equation

First impressions matter, even with equations! Looking at $-4+9.4(-2 x+1)=2(-7 x+1)+9.4$, we can see there are parentheses on both sides. This tells us that the distributive property will likely be our first step. We'll need to multiply the numbers outside the parentheses by the terms inside. Also, we notice the decimal number 9.4, which might seem intimidating, but don't worry, we'll handle it just like any other number. The key is to take it one step at a time and stay organized. A systematic approach is what separates mathematical masters from the mathematically-challenged, so let’s get our systematic thinking caps on, and jump right into it!

Key Principles for Solving Equations

Before we start crunching numbers, let's quickly review the fundamental principles that govern equation solving. The golden rule is: whatever you do to one side of the equation, you must do to the other side. This ensures that the equation remains balanced. We'll be using properties like the distributive property (a(b+c) = ab + ac), the commutative property (a+b = b+a), and the associative property (a+(b+c) = (a+b)+c) to manipulate the equation. Our aim is to simplify and isolate x using inverse operations. For instance, if we have addition, we'll use subtraction to undo it, and vice-versa. Similarly, for multiplication, we'll use division, and vice-versa. Keep these principles in mind as we proceed, and you'll be solving equations like a pro in no time!

Step-by-Step Solution

Alright, let's get our hands dirty and solve this equation! We'll break it down into manageable steps, so it's super easy to follow. Remember, the key is to be methodical and double-check each step to avoid any silly mistakes. Ready? Let's go!

Step 1: Apply the Distributive Property

As we spotted earlier, the first thing we need to tackle is those parentheses. The distributive property is our trusty tool here. It says that a(b + c) = ab + ac. So, let's apply it to both sides of our equation:

$-4+9.4(-2 x+1)=2(-7 x+1)+9.4$

On the left side, we distribute 9.4 to both -2x and 1:

$-4 + (9.4 * -2x) + (9.4 * 1) = 2(-7x + 1) + 9.4$

This simplifies to:

$-4 - 18.8x + 9.4 = 2(-7x + 1) + 9.4$

Now, let's do the same on the right side, distributing 2 to both -7x and 1:

$-4 - 18.8x + 9.4 = (2 * -7x) + (2 * 1) + 9.4$

Which simplifies to:

$-4 - 18.8x + 9.4 = -14x + 2 + 9.4$

Now our equation looks like this: $-4 - 18.8x + 9.4 = -14x + 2 + 9.4$. We've successfully eliminated the parentheses!

Step 2: Combine Like Terms

Next up, let's simplify each side of the equation by combining like terms. Like terms are those that have the same variable raised to the same power (or are just constants). On the left side, we have the constants -4 and 9.4. On the right side, we have the constants 2 and 9.4. Let's combine them:

Left side: -4 + 9.4 = 5.4

Right side: 2 + 9.4 = 11.4

So, our equation now looks cleaner:

$5.4 - 18.8x = -14x + 11.4$

See how much simpler it's becoming? Combining like terms helps us organize the equation and makes the next steps easier.

Step 3: Move the x Terms to One Side

Now, let's get all the x terms on one side of the equation. It doesn't matter which side we choose, but let's go for the left side in this case. To do this, we need to eliminate the -14x term on the right side. How do we do that? We use the inverse operation – addition! We'll add 14x to both sides of the equation:

$5.4 - 18.8x + 14x = -14x + 11.4 + 14x$

On the right side, -14x and +14x cancel each other out, leaving us with:

$5.4 - 18.8x + 14x = 11.4$

Now, let's combine the x terms on the left side: -18.8x + 14x = -4.8x

Our equation is now:

$5.4 - 4.8x = 11.4$

We're getting closer! All the x terms are on one side, and that's a big step.

Step 4: Move the Constants to the Other Side

Time to gather the constants on the other side of the equation. We want to isolate the x term, so we need to get rid of the 5.4 on the left side. We'll use the inverse operation again – subtraction. Let's subtract 5.4 from both sides:

$5.4 - 4.8x - 5.4 = 11.4 - 5.4$

On the left side, 5.4 and -5.4 cancel each other out, leaving us with:

$-4.8x = 11.4 - 5.4$

Now, let's simplify the right side: 11.4 - 5.4 = 6

Our equation is now:

$-4.8x = 6$

Look how simple it's become! We're just one step away from finding x.

Step 5: Isolate x by Dividing

Finally, the moment we've been waiting for – isolating x! We have -4.8 multiplied by x, so we need to use the inverse operation – division. We'll divide both sides of the equation by -4.8:

rac{-4.8x}{-4.8} = rac{6}{-4.8}

On the left side, -4.8 cancels out, leaving us with just x:

x = rac{6}{-4.8}

Now, let's simplify the right side. Dividing 6 by -4.8 gives us:

$x = -1.25$

Ta-da! We've solved for x. The value of x that satisfies the equation is -1.25.

Verification

Before we declare victory, it's always a good idea to verify our solution. This ensures we haven't made any mistakes along the way. To verify, we'll substitute our value of x (-1.25) back into the original equation and see if both sides are equal.

Original equation: $-4+9.4(-2 x+1)=2(-7 x+1)+9.4$

Substitute x = -1.25:

$-4 + 9.4(-2 * -1.25 + 1) = 2(-7 * -1.25 + 1) + 9.4$

Let's simplify each side:

Left side:

$-4 + 9.4(2.5 + 1) = -4 + 9.4(3.5) = -4 + 32.9 = 28.9$

Right side:

$2(8.75 + 1) + 9.4 = 2(9.75) + 9.4 = 19.5 + 9.4 = 28.9$

Both sides are equal! This confirms that our solution, x = -1.25, is correct. High five!

Conclusion

And there you have it, folks! We've successfully solved for x in the equation $-4+9.4(-2 x+1)=2(-7 x+1)+9.4$. We broke it down step-by-step, using the distributive property, combining like terms, and isolating x through inverse operations. Remember, the key to solving equations is to stay organized, follow the order of operations, and double-check your work. With practice, you'll become a master equation solver in no time. So, go forth and conquer those algebraic challenges! You've got this!