Solving Linear Equations: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into the world of linear equations. Solving them can seem tricky at first, but trust me, once you get the hang of it, it's like riding a bike. In this article, we'll break down the process of solving a linear equation, specifically the one we've got here: $3.4 + 2(9.7 - 4.8x) = 61.2$. We'll explore the crucial steps involved and why they're important. So, buckle up, grab your calculators (if you need them), and let's get started!

Understanding the Basics of Linear Equations

Before we jump into the equation itself, let's quickly recap what a linear equation is. Basically, it's an equation where the highest power of the variable (usually 'x') is 1. That means no x-squared, x-cubed, or anything fancy. These equations represent a straight line when graphed, hence the name 'linear'. The goal when solving a linear equation is to isolate the variable on one side of the equation. We do this by performing various algebraic operations, always remembering to keep the equation balanced. Any operation performed on one side MUST be performed on the other side as well. This is fundamental, guys. Think of it like a seesaw – to keep it balanced, any weight added or removed on one side must be mirrored on the other.

Okay, now that we have the fundamentals down, let's explore this particular equation. The initial equation, $3.4 + 2(9.7 - 4.8x) = 61.2$, might look a little intimidating with those parentheses, but don't worry. We'll break it down step by step. Remember, the key is to perform operations in the correct order to slowly chip away at the equation until you're left with 'x = something'. The correct approach involves several key steps. We'll get into the specifics in the following sections, but keep in mind that the primary goal is always to get 'x' by itself.

Now, let's consider the initial steps in solving this equation. The first thing you'll notice is the presence of parentheses. In algebra, parentheses indicate multiplication, and this is where we need to start. We will begin by focusing on the part of the equation within the parentheses. Following that, we will look to simplify the equation in a way that allows us to eventually isolate the variable. This will involve the use of operations like addition, subtraction, multiplication, and division, always taking care to maintain the equation's balance. This initial step sets the stage for the rest of the solution.

The Correct Steps to Solve the Equation

Now, let's get down to the nitty-gritty and figure out the correct steps to solve the linear equation: $3.4 + 2(9.7 - 4.8x) = 61.2$. Out of the available options, let's pinpoint the steps that will lead us to the correct solution. Remember, we need to choose four correct answers.

Step 1: Distribute the 2

This is where we begin the unravelling of the equation. The very first step is to distribute the 2 to both terms inside the parentheses. This means multiplying the '2' by both 9.7 and -4.8x. This is a critical step because it removes the parentheses and allows us to simplify the equation. The distribution step is all about getting rid of those parentheses and making the equation easier to work with. If we don’t distribute the 2, we will not be able to proceed.

So, when we distribute, the equation becomes: $3.4 + (2 * 9.7) - (2 * 4.8x) = 61.2$, which simplifies to $3.4 + 19.4 - 9.6x = 61.2$. Notice how the parentheses are gone, and now we have individual terms to work with. Remember to multiply each term inside the parentheses by 2. This step streamlines the equation and prepares it for further simplification. If we don't distribute, the equation won't make any sense. Always ensure you are multiplying by the entire term, including the sign.

Step 2: Combine the constants

After distributing, the next logical step is to combine like terms. In our case, we have two constant terms: 3.4 and 19.4. This involves adding these two numbers together. This simplifies the left side of the equation and reduces the number of terms we need to manage. Combining constants is a straightforward arithmetic operation, but it’s essential to keep the equation organized. Combining like terms is about simplifying the equation to make it easier to solve. When we combine 3.4 and 19.4, we get 22.8. Our equation now looks like this: $22.8 - 9.6x = 61.2$.

This simplifies the left side, bringing us one step closer to isolating 'x'. Always double-check your arithmetic, as a small error here can lead to an incorrect answer. This helps in streamlining the equation, bringing us one step closer to isolating the variable. Remember, the more simplified the equation is, the easier it becomes to solve.

Step 3: Subtract 22.8 from both sides

This is a crucial step in isolating 'x'. To get the term with 'x' by itself, we need to get rid of the constant term on the same side. We do this by subtracting 22.8 from both sides of the equation. Why both sides? Because we always need to maintain the balance of the equation, as we talked about earlier. Performing the same operation on both sides ensures that the equation remains valid. This ensures that the equation maintains its balance. This operation eliminates the constant term from the left side, leaving us with: $-9.6x = 38.4$.

Subtracting the value from both sides isolates the variable term. This action is pivotal to getting 'x' by itself. Ensure you perform the subtraction correctly on both sides to maintain the equation's integrity. It's a fundamental principle of algebra, and a solid understanding of it is necessary for solving linear equations. Remember, anything we do on one side, we must also do on the other side. This is like adding or subtracting equal weights from a scale; it keeps the scale balanced.

Step 4: Divide both sides by -9.6

We're in the final stretch now! We've isolated the 'x' term. The last step is to get 'x' completely alone. To do this, we need to divide both sides of the equation by the coefficient of 'x', which is -9.6. Remember, dividing by the coefficient is the opposite of multiplying by it. After dividing, we will have 'x' on one side and the result of the division on the other side. This isolates the variable completely, revealing its value.

So, the equation $-9.6x = 38.4$ becomes: $x = 38.4 / -9.6$. Thus, we find the solution: $x = -4$. That's it! We've solved the equation. Always double-check your calculations to ensure accuracy. Dividing both sides by the coefficient gives us the solution. It is the final step, and it is crucial to accurately calculating the result. And of course, double-check your calculation to ensure that you have it right. This step is about getting 'x' by itself, and once you have done that, you know your answer.

Conclusion

So there you have it, Plastik Magazine readers! Solving linear equations, like the one we tackled today, is all about following a series of logical steps. By distributing, combining like terms, and performing the correct operations on both sides of the equation, you can isolate the variable and find the solution. Remember to always double-check your work, and don't be afraid to practice. The more you practice, the easier it will become. Keep those algebraic skills sharp, and you'll be solving equations like a pro in no time! Remember, these skills are transferable to many aspects of your life. Keep practicing, and you'll do great! And that is how you solve linear equations. Now go solve some equations!