Solving $6.2 + 0.4x = 8.05$: A Step-by-Step Guide

Dec 19, 2025 by Alex Johnson 50 views

Welcome, aspiring mathematicians and curious minds! Today, we're diving into the exciting world of algebra to tackle a common challenge: solving linear equations. Specifically, we'll walk through how to solve the equation $6.2 + 0.4x = 8.05$ . Don't let the numbers with decimals intimidate you; by the end of this article, you'll feel confident in breaking down similar problems and finding that elusive 'x'. Understanding how to solve these equations is a fundamental skill, not just for higher mathematics, but for everyday problem-solving, budgeting, science, and even engineering. It's like learning the secret code to unlock countless real-world scenarios. We'll explore the core principles behind manipulating equations, ensuring that you grasp not just what to do, but why you're doing it. So, grab a pen and paper, and let's embark on this journey to master linear equations with a friendly, step-by-step approach. Our goal is to make the process of solving linear equations both clear and enjoyable, transforming what might seem complex into something entirely manageable. This particular problem, $6.2 + 0.4x = 8.05$ , is an excellent example of a basic algebraic equation that introduces you to working with decimals while applying the fundamental rules of algebra. By working through it diligently, you'll reinforce your understanding of isolating variables and maintaining the balance of an equation, which are crucial concepts in mathematics. This guide aims to provide comprehensive value, focusing on creating high-quality content that truly helps you learn and apply these vital mathematical skills. Whether you're a student struggling with homework or an adult looking to brush up on your algebra, this article is designed to be accessible and highly informative.

Understanding the Basics of Linear Equations

Before we jump directly into solving $6.2 + 0.4x = 8.05$ , let's first refresh our memory on what a linear equation actually is and the fundamental principles that govern its solution. A linear equation is essentially an algebraic equation in which each term has an exponent of 1, and the graph of which is a straight line. It typically involves a single variable (like 'x' in our case), constants (numbers without variables), and coefficients (numbers multiplied by variables). The entire purpose of solving a linear equation is to find the value of the unknown variable that makes the equation true. Think of an equation like a balanced scale: whatever you do to one side, you must do to the other side to keep it perfectly balanced. This concept is incredibly important and is often referred to as the properties of equality. There are four main properties of equality that we frequently use: the addition property of equality (you can add the same number to both sides), the subtraction property of equality (you can subtract the same number from both sides), the multiplication property of equality (you can multiply both sides by the same non-zero number), and the division property of equality (you can divide both sides by the same non-zero number). These properties are our primary tools for isolating the variable, which is our ultimate goal in solving any linear equation. When you face an equation like $6.2 + 0.4x = 8.05$ , your brain should immediately start thinking about how to move all the constant terms to one side of the equation and all the variable terms to the other. By understanding these foundational concepts, you'll approach the problem not just as a series of steps, but with a deeper comprehension of why each step is taken. This foundational knowledge is crucial for building confidence and developing strong problem-solving skills in algebra and beyond. It ensures that when you encounter more complex equations, you'll have a solid framework to rely upon, making the process of finding 'x' much more intuitive and less daunting. So, let's keep these fundamental rules in mind as we meticulously work through our specific equation, ensuring we maintain balance at every turn. We want to be sure that our understanding is rock-solid before we start moving numbers around and performing operations. The better your grasp of these basics, the more effortlessly you'll be able to solve similar algebraic problems in the future, proving that a strong foundation is the key to mathematical success.

Step-by-Step Guide: Solving $6.2 + 0.4x = 8.05$

Now, let's put those foundational concepts into practice and systematically solve our equation: $6.2 + 0.4x = 8.05$ . We'll break it down into manageable steps, making sure each operation is clear and understandable. The main objective here is to isolate the variable 'x' on one side of the equation. This means we want to get 'x' all by itself, with everything else on the other side. Think of it like carefully unwrapping a present; you remove the outer layers first to get to the desired item inside. For our equation, the 'outer layers' are the numbers that are added or subtracted from the term containing 'x'. Remember the balanced scale analogy: whatever we do to one side, we must do to the other. This ensures that the equality remains true throughout our entire solution process. We will tackle each step deliberately, explaining the reasoning behind every operation. This method not only helps in solving this specific problem but also builds a versatile framework for solving linear equations in general, regardless of how simple or complex they might initially appear. By the end of this section, you'll have a clear, actionable strategy for approaching similar algebraic challenges, making your journey through mathematics much smoother and more successful. This methodical approach is vital for building confidence and ensuring accuracy in your solutions. Let's get started!

Step 1: Isolate the Term with the Variable

Our first major step in solving $6.2 + 0.4x = 8.05$ is to isolate the term that contains our variable, 'x'. In this equation, that term is $0.4x$ . Currently, it's being added to a constant, $6.2$ . To get $0.4x$ by itself on the left side, we need to eliminate the $6.2$ . How do we do that? We use the subtraction property of equality. Since $6.2$ is a positive number, we can subtract $6.2$ from both sides of the equation. This maintains the balance of our equation and effectively moves the constant term to the right side. Let's see it in action:

Original equation:

$6.2 + 0.4x = 8.05$

Subtract $6.2$ from the left side:

$6.2 - 6.2 + 0.4x = 8.05$

To keep the balance, we must also subtract $6.2$ from the right side:

$6.2 - 6.2 + 0.4x = 8.05 - 6.2$

Now, perform the subtraction on both sides:

On the left side, $6.2 - 6.2$ becomes $0$ , leaving us with just $0.4x$ .

On the right side, we calculate $8.05 - 6.2$ . It's often helpful to align the decimal points when subtracting decimals:

  8.05
- 6.20
------
  1.85

So, $8.05 - 6.2$ equals $1.85$ .

Our equation now simplifies to:

$0.4x = 1.85$

See? We've successfully isolated the term with 'x'! This is a crucial step in the process of solving linear equations and brings us much closer to finding the value of 'x'. This operation is foundational and will be applied to nearly all linear equations you encounter. It’s about strategically moving numbers around while respecting the mathematical rules, making sure that at every stage, the equation remains perfectly valid and balanced. Understanding this step makes the whole process of solving $6.2 + 0.4x = 8.05$ much clearer, setting a solid foundation for the subsequent steps. This careful approach to isolating the variable term is a cornerstone of algebraic manipulation and is a skill you'll use repeatedly. Keep practicing this, and you'll find yourself able to handle even more complex equations with ease and confidence. The goal is always to simplify until 'x' stands alone.

Step 2: Isolate the Variable

Great job on isolating the term $0.4x$ ! Now, we're ready for the second major step in solving $6.2 + 0.4x = 8.05$ : getting 'x' completely by itself. Our current equation is $0.4x = 1.85$ . Here, 'x' is being multiplied by $0.4$ . To undo multiplication, we use its inverse operation: division. This means we'll apply the division property of equality. We need to divide both sides of the equation by $0.4$ . Remember, whatever you do to one side, you must do to the other to keep that mathematical scale balanced! This step is critical because it's the final push to reveal the exact value of our variable 'x'. By dividing both sides by the coefficient of 'x', we effectively cancel out that coefficient on the left side, leaving 'x' entirely on its own. It's a fundamental principle in algebraic problem-solving that allows us to find the unknown value.

Let's apply this:

Current equation:

$0.4x = 1.85$

Divide the left side by $0.4$ :

$rac{0.4x}{0.4} = 1.85$

To keep the balance, we must also divide the right side by $0.4$ :

$rac{0.4x}{0.4} = rac{1.85}{0.4}$

Now, perform the division on both sides:

On the left side, $rac{0.4x}{0.4}$ simplifies to $x$ , because any number divided by itself (except zero) is 1. So, $0.4/0.4 = 1$ , leaving $1x$ , which is simply $x$ .

On the right side, we need to calculate $rac{1.85}{0.4}$ . Dividing decimals can sometimes seem tricky, but a good strategy is to make the divisor (the number you're dividing by) a whole number. We can do this by multiplying both the numerator and the denominator by a power of 10. In this case, multiplying both by 10 will turn $0.4$ into $4$ and $1.85$ into $18.5$ :

$rac{1.85 imes 10}{0.4 imes 10} = rac{18.5}{4}$

Now, perform the long division:

So, $rac{1.85}{0.4}$ equals $4.625$ .

Therefore, our solution for x is:

$x = 4.625$

And there you have it! We've successfully isolated 'x' and found its value. This step concludes the primary mathematical operation of solving for the variable and is a critical point in our journey to master linear equations. Understanding this division process, especially with decimals, is a valuable skill in itself, reinforcing your overall numerical fluency. This demonstrates how even equations involving decimals can be systematically solved using fundamental algebraic principles. Knowing how to solve $6.2 + 0.4x = 8.05$ equips you with a powerful tool for tackling a wide array of mathematical problems.

Step 3: Verify Your Solution

We've found our answer for $x$ , which is $4.625$ . But how do we know if it's correct? The final and incredibly important step in solving $6.2 + 0.4x = 8.05$ (or any equation, for that matter!) is to verify your solution. This means substituting the value we found for 'x' back into the original equation and checking if both sides of the equation are equal. If they are, then our solution is correct! This step is often overlooked, but it's a powerful way to catch any errors and build confidence in your algebraic skills. It provides a self-check mechanism, ensuring that all your calculations and manipulations were done accurately. Think of it as double-checking your work before presenting it as final. It’s a habit that every great mathematician or problem-solver develops over time. By going through this verification process, you're not just confirming an answer; you're solidifying your understanding of how an equation functions and what it truly means for a value to be a