Solving The Linear Equation: -3/8x + 2 = 0

Hey guys! Let's dive into solving a linear equation today. Linear equations are fundamental in mathematics, and mastering them is super important for more advanced topics. In this article, we'll break down the steps to solve the equation $-\frac{3}{8}x + 2 = 0$. We’ll go through each step in detail, so you can understand not just the how, but also the why behind the solution. Ready to get started?

Understanding Linear Equations

Before we jump into solving our specific equation, let’s quickly recap what a linear equation actually is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called “linear” because when you graph them, they form a straight line. The general form of a linear equation is ax + b = 0, where x is the variable, and a and b are constants. Understanding this basic structure is crucial because it helps us approach different types of equations systematically.

In our case, the equation $-\frac{3}{8}x + 2 = 0$ fits this form perfectly. Here, a is $-\frac{3}{8}$ and b is 2. The goal when solving a linear equation is to isolate the variable x on one side of the equation. This means we want to manipulate the equation using algebraic operations until we have x = some value. This “some value” is the solution to the equation, which means it’s the value of x that makes the equation true. So, whenever you see a linear equation, think of it as a puzzle where your mission is to get x all by itself! Now that we've got the basics down, let's tackle our equation step by step. We'll start by identifying the key components and then move on to the actual solving process. Remember, each step we take is aimed at isolating x, making it easier to find its value. Understanding this goal will make the process much smoother and more intuitive.

Step-by-Step Solution

Okay, let's get down to business and solve the equation $-\frac{3}{8}x + 2 = 0$! We're going to take it one step at a time, so it's super clear how we arrive at the answer. Remember, our main goal here is to isolate x, which means getting it all by itself on one side of the equation.

Step 1: Isolate the Term with x

The first thing we want to do is get the term that includes x (which is $-\frac{3}{8}x$) by itself on one side of the equation. To do this, we need to get rid of the +2 that's hanging out there. The way we do that is by using the inverse operation. Since we have +2, we're going to subtract 2 from both sides of the equation. Remember, whatever you do to one side of the equation, you must do to the other side to keep things balanced. This is a golden rule in algebra! So, let's do it:

$-\frac{3}{8}x + 2 - 2 = 0 - 2$

This simplifies to:

$-\frac{3}{8}x = -2$

Awesome! We've now successfully isolated the term with x. This is a big step forward. Now we have a cleaner equation that's easier to work with. By subtracting 2 from both sides, we've maintained the equality while moving closer to our goal of isolating x. This step highlights the importance of performing the same operations on both sides of the equation to keep it balanced. Think of it like a scale: if you add or subtract weight from one side, you need to do the same on the other side to keep it level. This principle is fundamental in algebra and will help you solve all sorts of equations.

Step 2: Get Rid of the Fraction

Now we've got $-\frac{3}{8}x = -2$. The next hurdle is that pesky fraction in front of x. Fractions can sometimes make things look more complicated than they are, but don't worry, we've got a simple way to deal with it! The trick is to multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction is just the fraction flipped upside down. So, the reciprocal of $-\frac{3}{8}$ is $-\frac{8}{3}$.

Why does this work? When you multiply a fraction by its reciprocal, you get 1. And we want a 1 in front of our x because that means x is isolated! So, let's multiply both sides by $-\frac{8}{3}$:

$(-\frac{8}{3})(-\frac{3}{8}x) = (-2)(-\frac{8}{3})$

On the left side, the $-\frac{8}{3}$ and $-\frac{3}{8}$ cancel each other out, leaving us with just x. On the right side, we have $(-2)(-\frac{8}{3})$. Remember, when you multiply two negative numbers, you get a positive number. So, we have:

x = $\frac{16}{3}$

Fantastic! We've successfully gotten rid of the fraction and now x is all by itself. This step is a classic example of using inverse operations to simplify equations. Multiplying by the reciprocal is a super useful technique that you'll use again and again in algebra. It's like having a secret weapon against fractions! By understanding why this works – because a fraction times its reciprocal equals 1 – you're not just memorizing a step, you're learning a powerful mathematical concept.

Step 3: The Solution

Guess what? We've done it! We've successfully isolated x and found the solution to the equation. Our solution is:

x = $\frac{16}{3}$

That means that if you substitute $\frac{16}{3}$ for x in the original equation, it will be true. You can even double-check this by plugging it back into the original equation to make sure it works. This is always a good habit to get into, as it helps you catch any mistakes you might have made along the way. So, let's verify our solution:

$-\frac{3}{8}(\frac{16}{3}) + 2 = 0$

Simplifying, we get:

-2 + 2 = 0

0 = 0

Yep, it checks out! Our solution is correct. This final step of verifying the solution is crucial because it provides a double-check on our work. It's like the final piece of the puzzle clicking into place, confirming that we've solved the equation correctly. By substituting the value back into the original equation, we ensure that both sides are equal, validating our solution. This practice not only boosts confidence but also reinforces the understanding of the equation-solving process.

Conclusion

Alright guys, we did it! We successfully solved the linear equation $-\frac{3}{8}x + 2 = 0$ and found that x = $\frac{16}{3}$. We walked through each step, from isolating the term with x to getting rid of the fraction, and finally verifying our solution. Remember, the key to solving linear equations is to isolate the variable by using inverse operations and keeping the equation balanced.

Solving equations like this is a fundamental skill in math, and it's something you'll use again and again. So, make sure you understand each step and why it works. Keep practicing, and you'll become a pro at solving linear equations in no time! If you ever get stuck, just remember the basic principles: isolate the variable, use inverse operations, and always double-check your work. With these tools in your arsenal, you'll be able to tackle any linear equation that comes your way. Keep up the awesome work, and happy solving! Understanding linear equations opens the door to more complex mathematical concepts, so the effort you put in now will pay off big time later on.