Verifying (4, 0) As A Solution For X - 2y = 4 A Step-by-Step Guide
Introduction: Understanding Solutions to Linear Equations
In mathematics, particularly in algebra, linear equations play a foundational role. These equations, characterized by variables raised to the first power, represent straight lines when graphed on a coordinate plane. A solution to a linear equation is any ordered pair (x, y) that, when substituted into the equation, makes the equation a true statement. Verifying whether a given point is a solution is a fundamental skill in algebra, serving as a building block for more complex concepts. In this article, we will delve into the process of verifying whether the ordered pair (4, 0) is a solution to the linear equation x - 2y = 4. This exercise not only reinforces the concept of solutions but also provides a practical application of algebraic substitution. Understanding how to verify solutions is crucial for various mathematical tasks, such as solving systems of equations, graphing linear equations, and understanding the relationships between variables. Let's embark on this exploration to solidify our understanding of linear equations and their solutions.
This introduction sets the stage for a comprehensive understanding of linear equations and their solutions. The core concept of an ordered pair satisfying a linear equation is highlighted, emphasizing the importance of verifying solutions as a fundamental algebraic skill. The practical applications of this skill in more complex mathematical tasks are also mentioned, providing context for the exercise. By clearly defining the problem and its significance, the introduction aims to engage the reader and establish a solid foundation for the subsequent steps.
Moreover, the introduction emphasizes the role of algebraic substitution in verifying solutions, which is a key technique in algebra. This not only prepares the reader for the specific steps involved in the verification process but also underscores the broader applicability of substitution in mathematical problem-solving. The mention of graphing linear equations and understanding the relationships between variables further illustrates the connections between different algebraic concepts and the importance of mastering the basics. By providing a holistic view of the topic, the introduction encourages a deeper understanding and appreciation of the subject matter.
By carefully explaining the concept of a solution to a linear equation and its relevance, the introduction lays a strong foundation for the rest of the article. The reader is guided to understand the importance of verifying solutions as a fundamental algebraic skill, which is crucial for various mathematical tasks. The practical applications of this skill are also emphasized, providing context for the exercise. The introduction effectively sets the stage for a comprehensive understanding of linear equations and their solutions, ensuring that the reader is well-prepared for the subsequent steps in the verification process. This thoughtful approach to introducing the topic enhances the overall learning experience and fosters a deeper understanding of the subject matter.
Step-by-Step Verification Process
To verify whether (4, 0) is a solution to the equation x - 2y = 4, we will follow a systematic approach. The fundamental principle behind verifying solutions is substitution: we replace the variables in the equation with the corresponding values from the ordered pair. In this case, the ordered pair (4, 0) represents x = 4 and y = 0. The goal is to substitute these values into the equation and check if the left-hand side (LHS) of the equation equals the right-hand side (RHS). If LHS = RHS, then the ordered pair is indeed a solution; otherwise, it is not. This process is a cornerstone of algebra, allowing us to confirm whether a given set of values satisfies a particular equation.
The first step is to substitute the values of x and y into the equation. Replacing x with 4 and y with 0, we get: 4 - 2(0) = 4. This substitution transforms the equation into a numerical expression that we can evaluate. The subsequent step involves simplifying the expression. According to the order of operations (PEMDAS/BODMAS), we perform multiplication before subtraction. Therefore, we first calculate 2(0), which equals 0. Our equation now becomes: 4 - 0 = 4. This simplification reduces the equation to a more manageable form, making it easier to determine whether the equation holds true.
Next, we perform the subtraction: 4 - 0 equals 4. So, our equation simplifies to 4 = 4. This final step is the crucial verification: we compare the LHS and RHS of the equation. In this instance, the LHS (4) is equal to the RHS (4). This equality confirms that the ordered pair (4, 0) satisfies the equation x - 2y = 4. Therefore, we can confidently conclude that (4, 0) is a solution to the given linear equation. This step-by-step process not only demonstrates the verification but also reinforces the importance of following the correct order of operations and carefully substituting values. The clear and methodical approach ensures that the reader can easily follow along and understand the logic behind each step.
Detailed Explanation of Each Step
Let's break down the verification process into even more detail to ensure clarity and a thorough understanding. We begin with the given equation: x - 2y = 4. Our task is to determine if the ordered pair (4, 0) satisfies this equation. Remember, an ordered pair is written in the form (x, y), where the first value represents the x-coordinate and the second value represents the y-coordinate. In our case, x = 4 and y = 0. The core of the verification process lies in substituting these values into the original equation.
Substituting x = 4 and y = 0 into the equation x - 2y = 4 gives us: 4 - 2(0) = 4. This substitution is a critical step because it replaces the variables with specific numerical values, allowing us to evaluate the expression and determine if the equation holds true. Now, we need to simplify the left-hand side (LHS) of the equation, which is 4 - 2(0). To do this correctly, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These rules dictate the sequence in which mathematical operations should be performed.
According to PEMDAS/BODMAS, multiplication takes precedence over subtraction. Therefore, we must first multiply 2 by 0, which results in 0. Our equation now looks like this: 4 - 0 = 4. The next step is to perform the subtraction: 4 - 0 equals 4. So, the left-hand side of the equation simplifies to 4. Now, we have 4 = 4. The final step is to compare the simplified LHS with the right-hand side (RHS) of the original equation, which is also 4. Since 4 is indeed equal to 4, we can confidently conclude that the ordered pair (4, 0) satisfies the equation x - 2y = 4. This detailed explanation underscores the importance of each step and clarifies the underlying logic, ensuring that the reader fully grasps the verification process.
Conclusion: Confirming (4, 0) as a Solution
In conclusion, through a meticulous step-by-step verification process, we have confirmed that the ordered pair (4, 0) is indeed a solution to the linear equation x - 2y = 4. The process involved substituting the values of x and y from the ordered pair into the equation, simplifying the resulting expression, and verifying that the left-hand side (LHS) equals the right-hand side (RHS). This fundamental exercise highlights the importance of algebraic substitution and the order of operations in determining solutions to linear equations. Understanding how to verify solutions is crucial for various mathematical tasks, including solving systems of equations, graphing lines, and analyzing linear relationships.
This verification process serves as a foundational concept in algebra, providing a practical application of theoretical knowledge. By mastering this skill, students can confidently tackle more complex problems involving linear equations and systems of equations. The ability to verify solutions is not only essential for academic success but also for real-world applications where linear models are used to represent and solve problems. For instance, in economics, linear equations can model supply and demand relationships, and verifying solutions helps determine equilibrium points. In physics, linear equations can describe motion, and verifying solutions can confirm the accuracy of calculations.
Therefore, the exercise of verifying (4, 0) as a solution to x - 2y = 4 is more than just a mathematical exercise; it is a crucial step in developing algebraic proficiency and problem-solving skills. The detailed explanation of each step, from substitution to simplification and verification, ensures that the reader gains a thorough understanding of the process. This understanding empowers individuals to confidently approach similar problems and apply the same principles to more complex scenarios. By reinforcing the importance of attention to detail and logical reasoning, this exercise contributes to the development of critical thinking skills that are valuable in various aspects of life.
