System Of Equations Y=6x^2+1 And Y=x^2+4 Explained
We're presented with an intriguing system of equations, each representing a parabola:
The question posed challenges us to understand why this system boasts two distinct solutions. To unravel this, we must delve into the nature of these equations, their graphical representations, and the implications of their intersection points. Understanding the solutions of a system of equations is fundamental in mathematics, particularly when dealing with nonlinear equations like quadratics. Quadratic equations form the backbone of many real-world applications, from modeling projectile motion to designing parabolic reflectors. This exploration will not only address the specific question but also provide a broader understanding of how quadratic systems behave and how their solutions can be interpreted graphically and algebraically. We will dissect the given options, analyze the properties of parabolas, and ultimately determine the correct explanation for the existence of two solutions in this system. The process involves understanding the vertex form of a parabola, the concept of intersection points, and how these geometric interpretations translate into algebraic solutions. By the end of this discussion, you will have a robust grasp of the underlying principles and be able to tackle similar problems with confidence. The significance of this exercise extends beyond the classroom, as it hones critical thinking and problem-solving skills applicable to various fields.
Dissecting the Equations: Unveiling the Parabolas
Let's begin by examining each equation individually. The first equation, y = 6x² + 1, represents a parabola. We can recognize this immediately due to the presence of the x² term, which is the hallmark of a quadratic function. The coefficient '6' in front of the x² term dictates the parabola's vertical stretch – a larger coefficient means a narrower parabola. The '+ 1' at the end signifies a vertical shift, moving the entire parabola one unit upwards. This means the vertex of this parabola, its lowest point, is located at (0, 1). The second equation, y = x² + 4, also describes a parabola. Here, the coefficient of x² is 1, indicating a standard vertical stretch. The '+ 4' shifts this parabola four units upwards, placing its vertex at (0, 4). These observations are crucial. The different vertical shifts and stretches suggest that the two parabolas will indeed intersect, but the number of intersections remains to be determined. Understanding the properties of parabolas is key to visualizing the system's behavior. A parabola's vertex plays a pivotal role in determining its overall shape and position. The vertical shift directly impacts where the parabola sits on the y-axis, while the coefficient of the x² term dictates how quickly the parabola opens upwards. In this case, the first parabola is narrower and sits lower than the second parabola. This configuration sets the stage for potential intersection points. Analyzing the equations in this way allows us to move beyond abstract symbols and develop a concrete mental image of the parabolas and their relationship to each other. This visual approach is often invaluable when solving mathematical problems, as it provides intuition and helps to guide the algebraic steps. By understanding the individual components of each equation, we can build a stronger foundation for understanding the system as a whole.
Evaluating the Proposed Explanations
Now, let's turn our attention to the provided explanation options. Option A suggests that each graph has one vertex, which is a solution. While it's true that each parabola has a vertex, this statement misinterprets the concept of a solution to a system of equations. A solution isn't simply a characteristic point on one graph; it's a point that satisfies both equations simultaneously. In graphical terms, this means a point where the two graphs intersect. The vertex of one parabola might not lie on the other parabola, so it wouldn't be a solution to the system. This option highlights a common misconception about systems of equations and the meaning of a solution. It's important to emphasize that solutions are shared points, points that exist on all the graphs within the system. The concept of a solution as an intersection point is fundamental. A point lying only on one graph is not a solution to the system as a whole. Therefore, simply having a vertex doesn't qualify a point as a solution. This option serves as a valuable learning opportunity, reinforcing the core definition of a solution in the context of systems of equations. We must always remember that a solution must satisfy all equations in the system, not just one. Turning to option B, it proposes that the graphs of the equations intersect. This statement aligns perfectly with the graphical interpretation of solutions to a system of equations. Intersection points represent the locations where the two parabolas share the same x and y values, precisely what it means to be a solution. The question now becomes: why do these graphs intersect twice? This is where our earlier analysis of the parabolas' shapes and positions becomes crucial. The narrower parabola (y = 6x² + 1) sits lower and opens upwards more rapidly than the wider parabola (y = x² + 4). This configuration allows the wider parabola to intersect the narrower one at two distinct points, one on each side of the y-axis. Option B provides the correct framework for understanding the solution, but the key is to delve deeper into the reason for the intersections.
Why Two Solutions? Unraveling the Intersection
To solidify our understanding, let's explicitly find the solutions. We can do this by setting the two equations equal to each other, since at the points of intersection, the y-values must be the same:
6x² + 1 = x² + 4
Now, we can solve for x:
5x² = 3
x² = 3/5
x = ±√(3/5)
This reveals two distinct x-values, one positive and one negative. This confirms that there are indeed two intersection points. For each x-value, we can substitute back into either equation to find the corresponding y-value. For instance, if we use y = x² + 4:
For x = √(3/5):
y = (√(3/5))² + 4 = 3/5 + 4 = 23/5
For x = -√(3/5):
y = (-√(3/5))² + 4 = 3/5 + 4 = 23/5
Thus, the two solutions are (√(3/5), 23/5) and (-√(3/5), 23/5). This algebraic confirmation reinforces our graphical intuition. The two solutions arise because the parabolas intersect at two points, each with a unique x-coordinate but the same y-coordinate due to the symmetry of the parabolas. The difference in the vertical stretch and the vertical shifts creates the necessary conditions for these two intersections. The algebraic solution provides a precise verification of our graphical reasoning. The process of setting the equations equal to each other and solving for x is a standard technique for finding intersection points. The resulting quadratic equation highlights the fundamental connection between the algebraic representation and the geometric interpretation. By solving for x, we directly determine the x-coordinates of the intersection points. The subsequent substitution to find y completes the solution, giving us the full coordinates of the points where the parabolas meet. This methodical approach ensures that we not only understand why there are two solutions but also where those solutions are located.
The Power of Intersection: Visualizing Solutions
The key takeaway here is the power of visualizing solutions as intersections. Understanding the graphical representation of equations can often provide a much clearer understanding than simply manipulating algebraic symbols. In this case, picturing two parabolas with different shapes and positions makes it intuitively clear that they can intersect at two points. This visual intuition is a valuable tool in problem-solving. It allows us to anticipate the nature of solutions, to check the reasonableness of our algebraic results, and to gain a deeper appreciation for the connections between different mathematical concepts. The process of sketching the graphs, even roughly, can be immensely helpful. A quick sketch allows us to see the relative positions of the parabolas and to estimate the number of intersection points. This visual check can prevent errors and provide confidence in our solutions. Furthermore, the graphical approach highlights the concept of a solution as a shared point. It emphasizes that a solution is not an isolated value but rather a pair of values (x, y) that satisfies both equations. This understanding is crucial for tackling more complex systems of equations, where graphical methods may be the only practical way to visualize the solutions. The power of intersection extends beyond parabolas. It applies to any system of equations, regardless of the types of functions involved. The intersection points always represent the solutions, the points that simultaneously satisfy all equations in the system. This fundamental principle forms the bedrock of graphical problem-solving in mathematics.
Conclusion: Embracing Graphical Insights
In conclusion, the system of equations y = 6x² + 1 and y = x² + 4 has two solutions because the graphs of the equations, two parabolas, intersect at two distinct points. Option B accurately captures this fundamental reason. The narrower, lower parabola intersects the wider, higher parabola at two locations due to their differing shapes and vertical positions. This exploration underscores the importance of connecting algebraic equations with their graphical representations. Graphical insights can often provide a more intuitive understanding of mathematical concepts, leading to deeper comprehension and more effective problem-solving. By visualizing the parabolas and their intersection points, we can readily grasp why this system has two solutions. This approach not only answers the specific question but also equips us with a powerful tool for tackling a wide range of mathematical challenges. The ability to visualize mathematical relationships is a crucial skill, one that can transform abstract symbols into concrete images. This transformation unlocks a new level of understanding, making complex concepts more accessible and fostering a more intuitive approach to problem-solving. Embracing graphical insights allows us to move beyond rote memorization and develop a genuine appreciation for the beauty and interconnectedness of mathematics.
