Solving Equations -7.8x - 3.9 = 0 Step-by-Step Guide

In the realm of algebra, solving equations is a fundamental skill. This article provides a step-by-step guide on how to solve the equation -7.8x - 3.9 = 0. This is a linear equation, and understanding the process of solving it lays the foundation for tackling more complex algebraic problems. We will break down each step, ensuring clarity and comprehension for learners of all levels.

Understanding the Basics of Linear Equations

Before diving into the specifics of this equation, let's first grasp the basics of linear equations. 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 plotted on a graph, they form a straight line. The general form of a linear equation is ax + b = 0, where 'a' and 'b' are constants, and 'x' is the variable we need to solve for.

Solving a linear equation involves isolating the variable on one side of the equation. This is achieved by performing algebraic operations that maintain the equality of the equation. These operations include addition, subtraction, multiplication, and division. The key principle is to perform the same operation on both sides of the equation to ensure that the equation remains balanced. Understanding this principle is crucial for successfully solving any linear equation.

In our case, the equation -7.8x - 3.9 = 0 fits this general form, where a = -7.8 and b = -3.9. Our goal is to find the value of 'x' that satisfies this equation. This involves a series of algebraic manipulations that we will explore in detail in the following sections. Mastering the techniques for solving such equations is essential for various applications in mathematics, science, and engineering. The ability to solve linear equations provides a foundation for tackling more complex problems and understanding the relationships between different variables.

Step 1: Isolating the Term with the Variable

The first step in solving the equation -7.8x - 3.9 = 0 is to isolate the term containing the variable 'x'. In this case, the term is -7.8x. To isolate this term, we need to eliminate the constant term, which is -3.9. We can do this by adding 3.9 to both sides of the equation. This operation maintains the balance of the equation and moves us closer to our goal of isolating 'x'.

Adding 3.9 to both sides, we get:

-7.8x - 3.9 + 3.9 = 0 + 3.9

This simplifies to:

-7.8x = 3.9

Now, we have successfully isolated the term with the variable 'x' on the left side of the equation. This is a crucial step because it allows us to focus solely on solving for 'x' in the next stage. By adding the constant term to both sides, we have effectively moved it to the right side, leaving the term with 'x' by itself. This process is a fundamental technique in solving linear equations and is applicable to a wide range of algebraic problems. The next step will involve isolating 'x' completely by dealing with the coefficient -7.8.

Isolating the term with the variable is a common strategy in algebra, and understanding why it works is important. By performing the same operation on both sides, we ensure that the equality remains valid. This principle is the cornerstone of algebraic manipulation and allows us to rearrange equations to solve for unknown variables. In this instance, isolating -7.8x sets the stage for the final step of dividing both sides by the coefficient of 'x'.

Step 2: Solving for x by Division

After isolating the term -7.8x, the next step is to solve for 'x' itself. To do this, we need to eliminate the coefficient -7.8 that is multiplying 'x'. The inverse operation of multiplication is division, so we will divide both sides of the equation by -7.8. This will isolate 'x' on one side and give us its value.

Dividing both sides of the equation -7.8x = 3.9 by -7.8, we get:

(-7.8x) / -7.8 = 3.9 / -7.8

This simplifies to:

x = -0.5

Therefore, the solution to the equation -7.8x - 3.9 = 0 is x = -0.5. This means that if we substitute -0.5 for 'x' in the original equation, the equation will hold true. This final step completes the process of solving the linear equation and provides the value of the unknown variable.

Dividing both sides by the coefficient is a fundamental technique in algebra, and it's important to understand why it works. By performing the same operation on both sides, we maintain the balance of the equation. In this case, dividing by -7.8 effectively cancels out the coefficient, leaving 'x' isolated and allowing us to determine its value. This process is applicable to a wide range of linear equations and is a crucial skill for anyone studying algebra.

Step 3: Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the value we found for 'x' back into the original equation. This step helps to catch any potential errors made during the solving process and provides confidence in the correctness of the answer. In our case, we found that x = -0.5, so we will substitute this value into the equation -7.8x - 3.9 = 0.

Substituting x = -0.5 into the original equation, we get:

-7.8(-0.5) - 3.9 = 0

Now, we perform the multiplication:

3. 9 - 3.9 = 0

This simplifies to:

0 = 0

Since the equation holds true, our solution x = -0.5 is correct. This verification step confirms that we have successfully solved the equation and that the value we found for 'x' satisfies the original equation. Verification is a valuable step in any mathematical problem-solving process, especially in algebra. It not only helps to identify errors but also reinforces the understanding of the equation and the solution.

The process of verification highlights the importance of understanding the meaning of a solution. A solution to an equation is a value that, when substituted for the variable, makes the equation true. By substituting our solution back into the original equation and verifying that it holds true, we are essentially confirming that our value for 'x' is the one that satisfies the equation's conditions. This understanding is crucial for applying algebraic skills to real-world problems and for tackling more complex mathematical concepts.

Alternative Methods for Solving the Equation

While the step-by-step method described above is a standard approach to solving linear equations, there are alternative methods that can be used, especially as one gains more proficiency in algebra. These methods might offer different perspectives on the problem and can be useful in various situations. One such method involves rearranging the equation in a slightly different way before isolating the variable. For instance, we could first add 3.9 to both sides of the equation, as we did in Step 1. Then, instead of dividing directly, we could multiply both sides by a factor that eliminates the decimal in -7.8x. This approach might be preferred by some as it avoids immediate division with decimals.

Another method involves using graphical techniques. Linear equations can be represented as straight lines on a graph, and the solution to the equation corresponds to the point where the line intersects the x-axis. By plotting the graph of the equation y = -7.8x - 3.9, we can visually identify the x-intercept, which represents the solution to the equation. This method can be particularly helpful for visualizing the solution and understanding the relationship between the equation and its graphical representation. However, it might not be as precise as the algebraic method, especially when dealing with non-integer solutions.

These alternative methods highlight the flexibility of algebraic problem-solving and the importance of understanding the underlying principles. While the step-by-step method provides a clear and structured approach, exploring alternative methods can enhance one's problem-solving skills and provide a deeper understanding of the concepts. The choice of method often depends on personal preference and the specific characteristics of the equation. The key is to develop a strong foundation in the fundamental principles of algebra and to be able to apply them in various ways.

Common Mistakes to Avoid When Solving Equations

Solving equations is a fundamental skill in algebra, but it's also an area where common mistakes can occur. Being aware of these pitfalls can help learners avoid errors and improve their problem-solving accuracy. One of the most common mistakes is failing to perform the same operation on both sides of the equation. This violates the fundamental principle of maintaining equality and leads to incorrect solutions. For instance, if one adds a number to one side of the equation but forgets to add it to the other side, the equation becomes unbalanced, and the solution will be wrong.

Another frequent mistake is making errors in arithmetic, especially when dealing with negative numbers or decimals. Simple addition, subtraction, multiplication, or division errors can significantly impact the final answer. Therefore, it's crucial to double-check all calculations and pay close attention to signs and decimal places. In our equation, -7.8x - 3.9 = 0, a mistake in dividing 3.9 by -7.8 would lead to an incorrect value for 'x'.

Another common error is incorrectly applying the order of operations. When simplifying expressions, it's essential to follow the correct order (PEMDAS/BODMAS), which dictates that parentheses/brackets are handled first, followed by exponents/orders, then multiplication and division, and finally addition and subtraction. Ignoring this order can lead to incorrect simplification and, consequently, incorrect solutions. Additionally, students sometimes make mistakes in distributing negative signs or combining like terms. These errors can be avoided by carefully reviewing each step and paying attention to the details of the equation.

To minimize these mistakes, it's helpful to develop a systematic approach to problem-solving. This includes writing down each step clearly, double-checking calculations, and verifying the solution by substituting it back into the original equation. Practice is also crucial for developing accuracy and confidence in solving equations. By understanding common mistakes and taking steps to avoid them, learners can significantly improve their algebraic skills.

Conclusion

In conclusion, solving the equation -7.8x - 3.9 = 0 involves a series of algebraic steps aimed at isolating the variable 'x'. We began by understanding the basics of linear equations and the principle of maintaining equality. Then, we systematically isolated the term containing 'x' by adding 3.9 to both sides of the equation. Next, we solved for 'x' by dividing both sides by -7.8, which gave us the solution x = -0.5. We then verified this solution by substituting it back into the original equation, confirming its accuracy.

Throughout this process, we highlighted the importance of understanding the underlying principles of algebra and the need to perform operations consistently on both sides of the equation. We also discussed alternative methods for solving the equation, such as graphical techniques, and emphasized the flexibility of algebraic problem-solving. Furthermore, we addressed common mistakes to avoid, such as arithmetic errors and failing to follow the correct order of operations. By understanding these common pitfalls, learners can improve their accuracy and confidence in solving equations.

Solving linear equations is a fundamental skill in algebra and a building block for more advanced mathematical concepts. The step-by-step approach outlined in this article provides a clear and structured method for solving equations of this type. By mastering these techniques and practicing regularly, learners can develop strong algebraic skills and confidently tackle a wide range of mathematical problems. The ability to solve equations is not only essential for success in mathematics but also for various applications in science, engineering, and other fields. This comprehensive guide aims to equip readers with the knowledge and skills necessary to confidently solve equations and apply these skills to real-world scenarios.