Solving Systems Of Equations An In-Depth Analysis Of Y = -6x + 2 And -12x - 2y = -4

In this article, we delve into the intricacies of solving a system of linear equations. We are presented with the following system: $\begin{aligned} y & =-6 x+2 \\ -12 x-2 y & =-4 \end{aligned}$. Our mission is to determine the nature of the solution set for this system. To accomplish this, we will employ various techniques, including substitution, elimination, and graphical analysis. By carefully examining the equations, we can discern whether there exists a unique solution, infinitely many solutions, or no solution at all. This exploration will not only provide the answer but also enhance our understanding of linear systems and their behavior. Before diving into the solution, let's first understand the context and importance of solving such systems. Solving systems of linear equations is a fundamental concept in mathematics with wide-ranging applications in various fields, including engineering, economics, and computer science. These systems often arise when modeling real-world scenarios involving multiple variables and constraints. For instance, in economics, we might use a system of equations to determine the equilibrium price and quantity of goods in a market. In engineering, we might use them to analyze the forces acting on a structure or to design electrical circuits. Therefore, mastering the techniques for solving these systems is crucial for anyone pursuing a career in these fields. The ability to solve systems of equations is not just about finding numerical answers; it's about understanding the relationships between variables and how they interact. This understanding allows us to make informed decisions and predictions in a variety of contexts. Furthermore, the process of solving these systems often involves critical thinking and problem-solving skills that are valuable in any field. The system presented to us showcases the beauty and complexity that can arise in even simple mathematical structures. Let's embark on this mathematical journey and unravel the solution to this system of linear equations.

Method 1: Substitution

One effective method to solve this system is substitution. We can substitute the expression for y from the first equation into the second equation. The first equation is $y = -6x + 2$. Substituting this into the second equation, $-12x - 2y = -4$, we get $-12x - 2(-6x + 2) = -4$. Now, let's simplify this equation step-by-step. First, distribute the -2: $-12x + 12x - 4 = -4$. Combining like terms, we have $0x - 4 = -4$, which simplifies to $-4 = -4$. This result is a true statement, independent of the value of x. This implies that the two equations are dependent, meaning they represent the same line. Consequently, there are infinitely many solutions to this system. The substitution method provides a clear path to understanding the relationship between the equations. By substituting one equation into the other, we effectively eliminate one variable, allowing us to solve for the remaining variable. In this case, the elimination led to a true statement, which is a hallmark of systems with infinitely many solutions. This method highlights the power of algebraic manipulation in revealing the underlying structure of the system. However, the substitution method is not the only way to approach this problem. There are other techniques, such as elimination, which can provide alternative perspectives on the solution. Exploring different methods not only reinforces our understanding but also equips us with a versatile toolkit for tackling a variety of problems. The fact that the substitution method revealed an identity (-4 = -4) is a key indicator of the system's nature. It tells us that the equations are not independent and that the solution set is not a single point but an infinite set of points that lie on the same line. This is a crucial concept in linear algebra, and it's important to recognize the implications of such outcomes. Now, let's explore another method to confirm our findings and gain further insight into the system.

Method 2: Elimination

Another powerful technique for solving systems of equations is elimination. To apply this method, we aim to manipulate the equations so that when we add or subtract them, one of the variables is eliminated. Let's rewrite the given equations: $\begin{aligned} y & =-6 x+2 \\ -12 x-2 y & =-4 \end{aligned}$. To eliminate y, we can multiply the first equation by 2, resulting in $2y = -12x + 4$. Now, we can add this modified equation to the second equation: $\begin{aligned} 2y & =-12 x+4 \\ -2y & =12 x-4 \end{aligned}$. Adding these equations gives us $0 = 0$, which is a true statement regardless of the values of x and y. This confirms that the two equations are dependent and represent the same line. Therefore, there are infinitely many solutions. The elimination method provides a complementary perspective to the substitution method. By strategically manipulating the equations, we can directly eliminate one variable, simplifying the system and revealing its nature. In this case, the elimination process led to a fundamental identity (0 = 0), which solidifies our understanding that the system has infinitely many solutions. This method underscores the importance of recognizing patterns and relationships between equations. The ability to manipulate equations and identify opportunities for elimination is a valuable skill in mathematics and its applications. The fact that we arrived at the same conclusion using both substitution and elimination methods strengthens our confidence in the result. It demonstrates the robustness of these techniques and their ability to provide consistent insights into the behavior of linear systems. Moreover, the elimination method offers a more intuitive way to visualize the dependence between the equations. By adding the equations, we directly see that the variables cancel out, leaving us with an identity. This reinforces the idea that the equations are essentially different forms of the same equation. Now, let's consider a graphical approach to further illustrate the nature of the solution set.

Method 3: Graphical Analysis

A graphical approach provides a visual representation of the system of equations. Let's rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The first equation is already in this form: $y = -6x + 2$. The second equation, $-12x - 2y = -4$, can be rewritten as $-2y = 12x - 4$. Dividing both sides by -2, we get $y = -6x + 2$. Notice that both equations are identical. This means they represent the same line on the coordinate plane. When two lines coincide, every point on the line is a solution to the system. Thus, there are infinitely many solutions. The graphical method offers a powerful visual confirmation of our algebraic findings. By plotting the equations on a graph, we can directly observe their relationship and the nature of their intersection. In this case, the fact that the equations represent the same line is immediately apparent, reinforcing the concept of infinitely many solutions. This method is particularly useful for understanding the geometric interpretation of linear systems. It allows us to visualize the solution set as the intersection of the lines represented by the equations. When the lines coincide, the intersection is the entire line, indicating an infinite number of solutions. The graphical approach also highlights the limitations of relying solely on visual inspection. While it can provide a quick overview of the system, it may not always be accurate, especially when dealing with complex equations or systems with non-integer solutions. Therefore, it's essential to combine graphical analysis with algebraic methods to ensure a comprehensive understanding of the solution. Moreover, the graphical representation can be a valuable tool for communicating the solution to others. A visual representation can often convey the concept more effectively than a purely algebraic explanation. The fact that the two lines overlap completely emphasizes the dependence between the equations and the infinite nature of the solution set. In conclusion, the graphical method provides a valuable complement to the algebraic methods, offering a visual confirmation of our findings and enhancing our understanding of the system.

Conclusion

Through substitution, elimination, and graphical analysis, we've definitively determined that the system of equations $\begin{aligned} y & =-6 x+2 \\ -12 x-2 y & =-4 \end{aligned}$ has infinitely many solutions. This is because the two equations represent the same line. Option D, infinite number of solutions, is the correct answer. This exploration has not only provided the answer but has also demonstrated the interconnectedness of different problem-solving techniques in mathematics. By employing multiple methods, we have gained a deeper understanding of the system and its behavior. The ability to approach a problem from different angles is a valuable skill that can be applied to various mathematical challenges. Moreover, this example underscores the importance of recognizing the different types of solutions that can arise in linear systems. A system can have a unique solution, infinitely many solutions, or no solution at all. Understanding these possibilities and the conditions under which they occur is crucial for solving real-world problems modeled by linear equations. The fact that we encountered infinitely many solutions in this case highlights the importance of carefully analyzing the equations and their relationships. It also emphasizes the significance of interpreting the results in the context of the problem. The journey of solving this system has been a rewarding one, not just for the answer we obtained but for the insights we gained along the way. It serves as a reminder that mathematics is not just about finding solutions; it's about understanding the underlying concepts and the connections between them. This understanding empowers us to tackle more complex problems and to appreciate the beauty and elegance of mathematics. As we conclude this exploration, let's carry forward the lessons learned and apply them to future mathematical endeavors.