Solutions Of Linear Systems Analyzing Y = -6x + 2 And -12x - 2y = -4

When dealing with systems of linear equations, a fundamental question arises: how many solutions does the system possess? The answer reveals crucial information about the relationship between the equations and their graphical representations. In this article, we will delve into the linear system defined by the equations y = -6x + 2 and -12x - 2y = -4. We will explore different methods to determine whether the system has one solution, no solution, or an infinite number of solutions. This exploration will provide a comprehensive understanding of how to analyze linear systems and interpret their solutions effectively. Understanding the nature of solutions in linear systems is critical in various fields, including mathematics, physics, engineering, and economics, where such systems often model real-world phenomena. By mastering the techniques to analyze these systems, one can gain valuable insights into the underlying relationships between variables and make informed decisions based on these insights. In the context of this specific problem, we will begin by examining the structure of the given equations and then apply algebraic methods to identify the nature of their solution set. By meticulously working through each step, we aim to provide a clear and accessible explanation suitable for anyone seeking to understand the intricacies of solving linear systems. Ultimately, this article aims to empower readers with the knowledge and skills necessary to tackle similar problems with confidence and precision.

Rewriting the Equations: A Necessary First Step

The first step in analyzing the system is to rewrite the equations in a standard form, which facilitates comparison and manipulation. The first equation, y = -6x + 2, is already in slope-intercept form, which is convenient. However, the second equation, -12x - 2y = -4, can be simplified and rewritten to match the slope-intercept form as well. To do this, we will isolate y on one side of the equation. Starting with -12x - 2y = -4, we can add 12x to both sides, yielding -2y = 12x - 4. Next, we divide both sides by -2 to solve for y: y = -6x + 2. Now, it’s clear that both equations are y = -6x + 2. This transformation is critical because it immediately reveals that the two equations are not just similar but identical. Recognizing this identity is key to determining the number of solutions. This rewriting process not only simplifies the equations but also highlights their relationship, which is crucial for determining the solution set. By converting both equations to the same form, we eliminate any ambiguity and can clearly see whether the equations represent the same line, parallel lines, or intersecting lines. This step is often the most insightful, as it lays the groundwork for the rest of the analysis. Furthermore, this approach is universally applicable to any system of linear equations, making it a valuable tool in problem-solving.

Identifying Identical Equations: The Key to Infinite Solutions

Upon rewriting both equations, we observe that they are identical: y = -6x + 2. This means that both equations represent the same line on a graph. When two equations represent the same line, every point on the line is a solution to both equations. Therefore, the system has an infinite number of solutions. This is a fundamental concept in linear algebra: if two equations in a system are multiples of each other or, as in this case, exactly the same, they do not define distinct lines but rather the same line. The graphical interpretation of this is that the two lines coincide, meaning they overlap perfectly. As a result, any point that satisfies one equation will also satisfy the other, leading to an infinite solution set. This outcome contrasts sharply with systems that have either one unique solution (where lines intersect at a single point) or no solution (where lines are parallel and never intersect). Recognizing identical equations is thus a critical skill in solving linear systems, allowing for a quick and accurate determination of the solution type. In practical terms, this situation often arises when the equations describe the same constraint or condition from different perspectives, leading to redundant information in the system. Understanding these scenarios is essential for effective modeling and analysis in various mathematical and real-world applications.

Graphical Interpretation: Visualizing Infinite Solutions

To further illustrate why this system has an infinite number of solutions, consider the graphical representation. The equation y = -6x + 2 represents a straight line with a slope of -6 and a y-intercept of 2. When we graph this line, every point on it satisfies the equation. Since the second equation is identical to the first, it represents the exact same line. Therefore, when we plot both equations on the same coordinate plane, we see only one line. This single line encompasses all the points that satisfy both equations simultaneously. In graphical terms, the solution set of a system of equations is the set of points where the graphs of the equations intersect. In this case, since the lines are coincident, they intersect at every point along their length. This visual confirmation underscores the concept of infinite solutions: there are countless points that lie on this line, each representing a valid solution to the system. The graphical perspective provides an intuitive understanding of why identical equations lead to infinite solutions, reinforcing the algebraic findings. Moreover, this graphical method is a powerful tool for visualizing the solutions of linear systems in general, providing a complementary approach to algebraic methods. By sketching the graphs of the equations, one can quickly ascertain whether the system has a unique solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). This visual insight is invaluable for both problem-solving and conceptual understanding.

Eliminating Options: Why the Other Answers Are Incorrect

Now, let’s consider why the other answer choices are incorrect. Option A, “one solution: (0,0),” can be easily disproven by substituting the point (0,0) into the equations. For y = -6x + 2, substituting x = 0 and y = 0 yields 0 = -6(0) + 2, which simplifies to 0 = 2, a false statement. Therefore, (0,0) is not a solution. Option B, “one solution: (1,-4),” can be checked similarly. Substituting x = 1 and y = -4 into y = -6x + 2 gives -4 = -6(1) + 2, which simplifies to -4 = -4, a true statement. However, this does not mean that this is the only solution; it merely confirms that (1,-4) is one of the infinite solutions. Option C, “no solution,” is incorrect because we have already established that the equations represent the same line, meaning there are infinitely many solutions, not zero. This process of elimination is a valuable strategy in problem-solving, especially in multiple-choice questions. By understanding why certain options are incorrect, one can gain a deeper understanding of the problem itself. In this case, eliminating the incorrect options reinforces the understanding that identical equations lead to infinite solutions and highlights the importance of verifying solutions by substituting them back into the original equations. Furthermore, this approach underscores the significance of graphical interpretation in complementing algebraic methods, providing a holistic understanding of the solution landscape.

In conclusion, the linear system defined by y = -6x + 2 and -12x - 2y = -4 has an infinite number of solutions. This is because the two equations, upon simplification, are identical, representing the same line on a graph. This understanding is crucial in solving linear systems and interpreting their solutions. The methods used to arrive at this conclusion, including rewriting equations in standard form, recognizing identical equations, and understanding graphical interpretations, are fundamental tools in linear algebra. By mastering these techniques, one can confidently analyze and solve a wide range of linear systems, gaining valuable insights into the relationships between variables and their solutions. The exploration of this specific system serves as a model for approaching similar problems, emphasizing the importance of a systematic and comprehensive approach. Furthermore, this analysis underscores the interconnectedness of algebraic and graphical methods in mathematics, demonstrating how they can be used in tandem to deepen understanding and enhance problem-solving skills. Ultimately, this article aims to equip readers with the knowledge and skills necessary to tackle linear systems with confidence and precision, paving the way for further exploration of mathematical concepts and their applications in various fields.