Solving 0.9x - 7.4 = -0.4x + 4.3 A Step-by-Step Algebraic Solution
Algebra problems often involve solving equations, and one common type is the linear equation. This article provides a detailed walkthrough of how to solve the linear equation 0.9x - 7.4 = -0.4x + 4.3. Understanding the process of solving linear equations is a fundamental skill in algebra, applicable in various mathematical and real-world contexts. This step-by-step guide will not only provide the solution but also explain the underlying principles, ensuring a comprehensive understanding of the method. By the end of this guide, you'll be equipped with the knowledge and skills to tackle similar algebraic problems with confidence. Let’s dive into the process of solving this equation, breaking down each step for clarity and ease of understanding. This method will help you not only solve this particular equation but also provide you with a framework for approaching other linear equations you may encounter. We’ll focus on the logic behind each step, making sure you grasp the concepts rather than just memorizing the procedures. Remember, algebra is built upon a foundation of logical steps, and mastering these fundamentals will open doors to more complex mathematical concepts. Linear equations are the building blocks of algebra, and with a solid grasp of how to solve them, you'll be well-prepared for more advanced topics. This detailed explanation is designed to make the process accessible and understandable for learners of all levels, whether you're a student just starting out in algebra or someone looking to refresh your skills. Understanding how to manipulate equations to isolate variables is a crucial skill in mathematics, and this guide will help you master that art.
1. Understanding the Equation
The linear equation we need to solve is 0.9x - 7.4 = -0.4x + 4.3. This equation involves a single variable, 'x', and our goal is to isolate 'x' on one side of the equation to find its value. Linear equations are characterized by the fact that the variable is raised to the power of 1, which makes them relatively straightforward to solve using algebraic manipulations. To begin, we must recognize the structure of the equation: it has two sides, separated by an equals sign (=). Each side contains terms involving 'x' and constant terms (numbers without a variable). The key to solving this equation is to perform operations that maintain the equality while moving terms around to isolate 'x'. These operations must be applied to both sides of the equation to ensure that the equation remains balanced. This principle of maintaining balance is fundamental to solving any algebraic equation. By understanding this basic structure, we can develop a strategic approach to isolate 'x'. The left side of the equation (0.9x - 7.4) contains a term with 'x' (0.9x) and a constant term (-7.4). Similarly, the right side of the equation (-0.4x + 4.3) contains a term with 'x' (-0.4x) and a constant term (4.3). Our strategy will involve moving the 'x' terms to one side and the constant terms to the other side. This process requires careful application of algebraic principles, particularly the idea that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain balance. Identifying the terms and their positions is the first crucial step in developing an effective solution strategy. This sets the stage for the subsequent steps, where we will apply algebraic operations to isolate the variable and find its value. A clear understanding of the equation's structure allows us to approach the solution process with a systematic and logical mindset.
2. Combining 'x' Terms
The next step in solving the equation 0.9x - 7.4 = -0.4x + 4.3 is to combine the 'x' terms on one side. To do this, we can add 0.4x to both sides of the equation. This operation will eliminate the -0.4x term on the right side and move it to the left side, combining it with the existing 0.9x term. Adding 0.4x to both sides ensures that we maintain the equality of the equation, a crucial principle in algebra. The equation now becomes 0.9x + 0.4x - 7.4 = -0.4x + 0.4x + 4.3. Simplifying this, we get 1.3x - 7.4 = 4.3. This step is essential because it consolidates all the terms involving 'x' into a single term, making the equation simpler to manage. By combining like terms, we reduce the complexity of the equation and bring ourselves closer to isolating the variable 'x'. This process is a standard technique in solving linear equations and is based on the commutative and associative properties of addition. These properties allow us to rearrange and group terms without changing the equation's value. The goal of combining 'x' terms is to make the equation easier to manipulate in subsequent steps. A clear understanding of this process is vital for students learning algebra, as it forms the basis for solving more complex equations. The ability to combine like terms efficiently and accurately is a fundamental skill that can be applied in various mathematical contexts. This step also highlights the importance of performing the same operation on both sides of the equation to maintain balance. By adding 0.4x to both sides, we ensured that the equation remained valid, and we were able to simplify it effectively. This step sets the stage for isolating 'x' in the following steps, bringing us closer to the final solution. Combining 'x' terms is not just a procedural step; it is a logical simplification that is essential for solving linear equations.
3. Isolating the 'x' Term
With the equation now in the form 1.3x - 7.4 = 4.3, our next goal is to isolate the term containing 'x'. This means we need to get 1.3x by itself on one side of the equation. To achieve this, we add 7.4 to both sides. This operation will cancel out the -7.4 on the left side, leaving us with just the 1.3x term. Adding 7.4 to both sides maintains the balance of the equation, a principle that is fundamental in algebraic manipulations. The equation becomes 1.3x - 7.4 + 7.4 = 4.3 + 7.4. Simplifying this, we get 1.3x = 11.7. This step is crucial because it brings us closer to solving for 'x' by eliminating the constant term from the left side of the equation. By isolating the term containing 'x', we set up the final step of dividing to find the value of 'x'. This process is a standard technique in solving linear equations and is based on the properties of equality. The goal of isolating the 'x' term is to simplify the equation so that we can easily solve for the variable. A clear understanding of this process is vital for students learning algebra, as it forms the basis for solving a wide range of equations. The ability to isolate terms efficiently and accurately is a fundamental skill that can be applied in various mathematical contexts. This step also highlights the importance of performing the same operation on both sides of the equation to maintain balance. By adding 7.4 to both sides, we ensured that the equation remained valid, and we were able to isolate the 'x' term effectively. This sets the stage for the final step, where we will divide to find the value of 'x'. Isolating the 'x' term is not just a procedural step; it is a logical simplification that is essential for solving linear equations.
4. Solving for 'x'
Now that we have the equation 1.3x = 11.7, the final step is to solve for 'x'. To do this, we divide both sides of the equation by 1.3. This operation will isolate 'x' on the left side and give us its value. Dividing both sides by 1.3 maintains the balance of the equation, ensuring that our solution is accurate. The equation becomes (1.3x) / 1.3 = 11.7 / 1.3. Simplifying this, we get x = 9. This is the solution to the equation. We have successfully isolated 'x' and found its value by performing a series of algebraic operations. This process is a fundamental technique in solving linear equations and is based on the properties of equality. The goal of solving for 'x' is to find the value that makes the equation true. A clear understanding of this process is vital for students learning algebra, as it forms the basis for solving a wide range of equations. The ability to solve for variables efficiently and accurately is a fundamental skill that can be applied in various mathematical contexts. This step also highlights the importance of performing the same operation on both sides of the equation to maintain balance. By dividing both sides by 1.3, we ensured that the equation remained valid, and we were able to isolate 'x' effectively. This final step completes the solution process, providing us with the value of 'x'. Solving for 'x' is not just a procedural step; it is the culmination of all the previous steps, leading to the answer. The solution x = 9 means that if we substitute 9 for 'x' in the original equation, both sides of the equation will be equal. This confirms that our solution is correct and demonstrates the power of algebraic manipulation in solving equations.
5. Verifying the Solution
To ensure the accuracy of our solution, it's essential to verify it. We substitute the value of x, which we found to be 9, back into the original equation: 0.9x - 7.4 = -0.4x + 4.3. This verification step is a crucial part of the problem-solving process in algebra, as it helps to catch any potential errors made during the solution steps. By substituting the solution back into the original equation, we can confirm whether the left side of the equation equals the right side, thus validating our solution. If the two sides do not equate, it indicates that there might be an error in our calculations, and we need to revisit the steps to identify and correct the mistake. This process reinforces the importance of careful and systematic work in algebra. Substituting x = 9, the equation becomes 0.9(9) - 7.4 = -0.4(9) + 4.3. Now, we simplify both sides of the equation separately. On the left side, 0.9(9) equals 8.1, so the left side becomes 8.1 - 7.4, which equals 0.7. On the right side, -0.4(9) equals -3.6, so the right side becomes -3.6 + 4.3, which also equals 0.7. Since both sides of the equation are equal (0.7 = 0.7), our solution x = 9 is correct. This verification step not only confirms the correctness of our solution but also reinforces our understanding of the equation and the algebraic manipulations we performed. It provides a sense of confidence in our problem-solving abilities and demonstrates the importance of checking our work. Verifying the solution is a standard practice in algebra and is highly recommended for students and practitioners alike. It ensures that we have not made any mistakes and that our answer is indeed the correct one. This step underscores the importance of accuracy in mathematical calculations and the need to double-check our work before finalizing a solution. By verifying our solution, we complete the problem-solving process and ensure that we have a reliable answer.
Conclusion
In conclusion, solving the linear equation 0.9x - 7.4 = -0.4x + 4.3 involves a series of algebraic steps designed to isolate the variable 'x'. The process includes combining like terms, moving terms across the equals sign while maintaining the equation's balance, and finally, dividing to find the value of 'x'. The step-by-step method explained in this guide provides a clear and systematic approach to solving such equations. Starting with an understanding of the equation's structure, we combined the 'x' terms on one side and the constant terms on the other. We then isolated the 'x' term and solved for 'x' by dividing both sides by the coefficient of 'x'. Finally, we verified our solution by substituting it back into the original equation to ensure its accuracy. This methodical approach is not only effective for solving this particular equation but also applicable to a wide range of linear equations. The key to success in algebra lies in understanding the fundamental principles and applying them consistently. The principles of equality, such as adding, subtracting, multiplying, or dividing both sides by the same value, are crucial for maintaining the balance of the equation and arriving at the correct solution. The importance of verifying the solution cannot be overstated, as it provides a check against potential errors and reinforces the understanding of the problem. By following these steps and principles, anyone can confidently tackle linear equations and build a strong foundation in algebra. The skills acquired in solving linear equations are essential for further studies in mathematics and have practical applications in various real-world scenarios. Mastering these skills empowers individuals to approach mathematical problems with confidence and accuracy. The ability to solve equations is a valuable asset in various fields, including science, engineering, finance, and technology. This comprehensive guide aims to equip learners with the necessary tools and knowledge to solve linear equations effectively and efficiently. The process of solving equations is not just about finding the right answer; it is also about developing critical thinking and problem-solving skills. This guide encourages a deep understanding of the underlying concepts, rather than just memorizing steps, to foster a lifelong appreciation for mathematics.
