Identifying Solutions For Graphed System Of Equations Y=x^2+x-2 And Y=2x-2

In the realm of mathematics, particularly in algebra and graphical analysis, systems of equations play a pivotal role in modeling real-world scenarios and solving complex problems. A system of equations comprises two or more equations with shared variables, and the solution to such a system is the set of values that satisfy all equations simultaneously. Graphically, this solution is represented by the point(s) where the graphs of the equations intersect. In this article, we delve into the intricacies of identifying solutions for a graphed system of equations, specifically focusing on a quadratic equation and a linear equation. We will explore the fundamental concepts, graphical interpretations, and algebraic methods to determine the points of intersection, thereby representing the solution(s) of the system. Understanding these solutions is crucial as they provide valuable insights into the relationships between the equations and their applications in various fields, including physics, engineering, and economics. Let's embark on this mathematical journey to unravel the significance of graphed systems and their solutions.

The graphed system of equations under consideration involves two distinct types of equations: a quadratic equation and a linear equation. The quadratic equation is given by y = x^2 + x - 2, which represents a parabola when plotted on a coordinate plane. The parabola is a U-shaped curve that opens upwards due to the positive coefficient of the x^2 term. The linear equation, y = 2x - 2, represents a straight line with a slope of 2 and a y-intercept of -2. Graphically, the solution(s) to this system of equations correspond to the point(s) where the parabola and the line intersect. These intersection points are crucial because they represent the x and y values that satisfy both equations simultaneously. To find these solutions, we can employ both graphical and algebraic methods. The graphical approach involves plotting both equations on the same coordinate plane and visually identifying the points of intersection. The algebraic approach, on the other hand, involves solving the equations simultaneously by substitution or elimination methods. In this article, we will explore both methods to provide a comprehensive understanding of how to determine the solutions of the graphed system.

The graphical interpretation of solutions to a system of equations is a powerful tool for visualizing the relationships between the equations. When we graph the quadratic equation y = x^2 + x - 2 and the linear equation y = 2x - 2 on the same coordinate plane, the solutions are represented by the points where the two graphs intersect. The parabola y = x^2 + x - 2 has roots at x = -2 and x = 1, which can be found by setting y = 0 and solving the quadratic equation. The vertex of the parabola, which is the minimum point, can be found using the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. In this case, the vertex is at x = -1/2, and the corresponding y value is y = (-1/2)^2 + (-1/2) - 2 = -9/4. The linear equation y = 2x - 2 has a y-intercept at (0, -2) and a slope of 2. By plotting these two graphs, we can visually identify the points of intersection. These points represent the x and y values that satisfy both equations. For instance, if the graphs intersect at (1, 0), this means that when x = 1, both equations yield y = 0. Similarly, any other intersection point represents another solution to the system. The accuracy of this graphical method depends on the precision of the graph, but it provides a clear visual representation of the solutions.

To find the solutions of the system of equations y = x^2 + x - 2 and y = 2x - 2 algebraically, we can use the method of substitution. Since both equations are expressed in terms of y, we can set them equal to each other: x^2 + x - 2 = 2x - 2. This creates a single equation in terms of x, which we can solve to find the x-coordinates of the intersection points. Subtracting 2x - 2 from both sides, we get x^2 + x - 2 - (2x - 2) = 0, which simplifies to x^2 - x = 0. Factoring out x, we have x(x - 1) = 0. This equation has two solutions: x = 0 and x = 1. These are the x-coordinates of the intersection points. To find the corresponding y-coordinates, we can substitute these x-values into either of the original equations. Using the linear equation y = 2x - 2, when x = 0, y = 2(0) - 2 = -2, and when x = 1, y = 2(1) - 2 = 0. Thus, the solutions to the system of equations are the points (0, -2) and (1, 0). These points are where the parabola and the line intersect on the graph, confirming our graphical interpretation. This algebraic method provides a precise way to find the solutions, ensuring accuracy and eliminating the potential for visual estimation errors.

The question presents two options for the solutions of the graphed system of equations: A) (-2,0) and (0,1), and B) (0,-2) and (1,0). To determine which option correctly represents the solutions, we need to verify which points satisfy both equations y = x^2 + x - 2 and y = 2x - 2. Let's analyze each point in both options. For option A, the points are (-2,0) and (0,1). For the point (-2,0), substituting into the quadratic equation gives 0 = (-2)^2 + (-2) - 2 = 4 - 2 - 2 = 0, which is true. Substituting into the linear equation gives 0 = 2(-2) - 2 = -4 - 2 = -6, which is false. Therefore, (-2,0) is not a solution. For the point (0,1), substituting into the quadratic equation gives 1 = (0)^2 + (0) - 2 = -2, which is false. Thus, (0,1) is not a solution either. Option A is incorrect. Now, let's analyze option B, which presents the points (0,-2) and (1,0). For the point (0,-2), substituting into the quadratic equation gives -2 = (0)^2 + (0) - 2 = -2, which is true. Substituting into the linear equation gives -2 = 2(0) - 2 = -2, which is also true. Thus, (0,-2) is a solution. For the point (1,0), substituting into the quadratic equation gives 0 = (1)^2 + (1) - 2 = 1 + 1 - 2 = 0, which is true. Substituting into the linear equation gives 0 = 2(1) - 2 = 2 - 2 = 0, which is also true. Therefore, (1,0) is a solution. Option B correctly represents the solutions of the graphed system of equations.

In conclusion, determining the solutions of a graphed system of equations involves understanding the graphical and algebraic relationships between the equations. For the system consisting of the quadratic equation y = x^2 + x - 2 and the linear equation y = 2x - 2, the solutions are the points where the parabola and the line intersect. Graphically, these points can be visualized on a coordinate plane, providing a clear representation of the solutions. Algebraically, the solutions can be found by setting the equations equal to each other and solving for x, then substituting the x-values back into either equation to find the corresponding y-values. In this case, the solutions were determined to be (0, -2) and (1, 0). By analyzing the given options, we verified that option B, which presented these points, correctly represented the solutions of the system. This process highlights the importance of both graphical interpretation and algebraic methods in solving systems of equations. Understanding these concepts is fundamental in various mathematical applications and real-world problem-solving scenarios. The ability to identify solutions for graphed systems equips us with a powerful tool for analyzing and interpreting complex relationships between equations, making it a crucial skill in mathematics and beyond.