What Are The Solution(s) Of The System Of Equations Represented By The Graphs Of $y = X^2 + X - 2$ And $y = 2x - 2$?

by ADMIN 117 views

Introduction

In the realm of mathematics, solving systems of equations is a fundamental skill. It allows us to find the point(s) where two or more equations intersect, representing shared solutions. When dealing with graphed systems of equations, this translates to identifying the points where the graphs of the equations intersect. In this article, we will delve into the process of finding the solution(s) of the graphed system of equations y = x^2 + x - 2 and y = 2x - 2. We will explore the graphical representation of these equations, discuss the concept of intersection points, and ultimately identify the correct solution(s).

Understanding the Equations

Before we dive into the solution, let's first understand the individual equations we are dealing with.

  • y = x^2 + x - 2: This equation represents a parabola, a U-shaped curve. The x^2 term indicates that it is a quadratic equation, and the coefficients determine the shape and position of the parabola. To visualize this equation, we can plot points by substituting different values of x and calculating the corresponding y values. We will find that the parabola opens upwards and intersects the x-axis at two points.
  • y = 2x - 2: This equation represents a straight line. The 2x term indicates the slope of the line, and the -2 represents the y-intercept, the point where the line crosses the y-axis. To visualize this equation, we can again plot points or use the slope-intercept form to draw the line. We will find that the line has a positive slope and intersects the y-axis at -2.

Graphical Representation and Intersection Points

To solve the system of equations graphically, we need to plot both equations on the same coordinate plane. The points where the graphs of the two equations intersect represent the solutions to the system. These intersection points are the ordered pairs (x, y) that satisfy both equations simultaneously.

Imagine plotting the parabola y = x^2 + x - 2 and the line y = 2x - 2 on a graph. The parabola will curve upwards, and the line will have a positive slope. The points where these two graphs cross each other are the solutions we are looking for. These points represent the x and y values that make both equations true.

Finding the Solution(s)

To find the solution(s) graphically, we can visually inspect the graph and identify the intersection points. However, for a more precise solution, we can use algebraic methods.

One way to solve the system algebraically is by setting the two equations equal to each other: x^2 + x - 2 = 2x - 2. This is because at the intersection points, the y-values of both equations are the same. Now we have a single equation in terms of x, which we can solve.

Subtracting 2x and adding 2 to both sides, we get: x^2 - x = 0. Factoring out an x, we have: x(x - 1) = 0. This equation has two solutions: x = 0 and x = 1.

These x-values represent the x-coordinates of the intersection points. To find the corresponding y-coordinates, we can substitute these x-values back into either of the original equations. Let's use the simpler equation, y = 2x - 2.

For x = 0, we get y = 2(0) - 2 = -2. So, one solution is (0, -2).

For x = 1, we get y = 2(1) - 2 = 0. So, the other solution is (1, 0).

Therefore, the solutions to the graphed system of equations y = x^2 + x - 2 and y = 2x - 2 are (0, -2) and (1, 0).

Analyzing the Options

The question presents an option A. (-2, 0] and (0, 1]. Comparing our calculated solutions with the provided option, we can see that the solution set does not match. Our solutions are the distinct points (0, -2) and (1, 0), whereas option A suggests intervals or a range of values, which is not accurate in this case.

Therefore, option A is not correct.

Why Graphical Solutions Matter

Graphical solutions provide a visual representation of the relationship between equations and their solutions. They allow us to see the intersection points, which represent the values that satisfy all equations in the system. This visual understanding can be particularly helpful when dealing with more complex equations or systems with multiple solutions.

Conclusion

In conclusion, to find the solution(s) of a graphed system of equations, we need to identify the points where the graphs of the equations intersect. These intersection points represent the ordered pairs (x, y) that satisfy all equations in the system. For the system y = x^2 + x - 2 and y = 2x - 2, we found the solutions to be (0, -2) and (1, 0) by both graphical and algebraic methods. These solutions represent the points where the parabola and the line intersect on the coordinate plane. Understanding how to solve systems of equations graphically is a valuable skill in mathematics, providing a visual and intuitive approach to finding solutions.

Deep Dive into Solving Systems of Equations Graphically

When we encounter a system of equations, we're essentially looking for the values that satisfy all the equations simultaneously. In the realm of graphical solutions, this translates to finding the points where the graphs of the equations intersect. This article will provide a comprehensive exploration of solving systems of equations graphically, with a particular focus on the example system: y = x^2 + x - 2 and y = 2x - 2. We'll dissect each equation, understand their graphical representations, delve into the concept of intersection points, and meticulously identify the accurate solution(s).

Deconstructing the Equations

Before embarking on the graphical journey, it's crucial to grasp the nature of each equation individually. This understanding lays the foundation for interpreting their graphical behavior and eventual intersection.

  • The Parabola: y = x^2 + x - 2

    This equation immediately signals a parabola, a graceful U-shaped curve. The presence of the x^2 term is the telltale sign of a quadratic equation, and the coefficients diligently orchestrate the parabola's shape and position on the coordinate plane. To truly visualize this equation, we can embark on a point-plotting expedition. By strategically substituting various values for x and diligently calculating the corresponding y values, we can map out the parabola's trajectory. The resulting curve will gracefully open upwards, a consequence of the positive coefficient of the x^2 term. Furthermore, we'll discover that this parabola intersects the x-axis at two distinct points, revealing its roots or zeros. The vertex of the parabola, the point where the curve changes direction, is also a crucial characteristic to consider.

  • The Straight Line: y = 2x - 2

    In stark contrast to the parabola, this equation proudly proclaims itself as a straight line. The term 2x dictates the line's slope, indicating its steepness and direction. The constant term, -2, boldly declares the y-intercept, the precise point where the line elegantly crosses the y-axis. To visualize this linear equation, we can once again employ the point-plotting technique, but the slope-intercept form (y = mx + b) offers an even more efficient approach. The slope, m, guides us on how much the line rises or falls for every unit increase in x, and the y-intercept, b, anchors the line's position on the vertical axis. This line, with its positive slope, will ascend gracefully from left to right, intersecting the y-axis at the point (0, -2).

Graphical Harmony: Intersection Points

The essence of solving a system of equations graphically lies in the harmonious dance of the individual graphs. When we plot both equations on the same coordinate plane, we create a visual symphony of curves and lines. The points where these graphs intersect are the coveted solutions to the system. These intersection points, expressed as ordered pairs (x, y), represent the values that simultaneously satisfy both equations. They are the shared coordinates that reside on both the parabola and the line.

Imagine the parabola from y = x^2 + x - 2 gracefully curving upwards and the line from y = 2x - 2 elegantly slicing through the plane. The points where these two entities meet are the solutions we seek. Each point is a testament to the shared values of x and y that make both equations ring true.

Unveiling the Solutions: A Step-by-Step Approach

To pinpoint the solutions graphically, we can initially rely on visual inspection of the graph. However, for unwavering precision, algebraic methods become our trusted allies. One powerful technique is to equate the two equations, recognizing that at the intersection points, the y-values must be identical.

So, we boldly declare: x^2 + x - 2 = 2x - 2. This creates a single equation, now solely in terms of x, which we can skillfully solve.

Through meticulous algebraic manipulation, we subtract 2x and add 2 to both sides, leading us to: x^2 - x = 0. The art of factoring then reveals: x(x - 1) = 0. This equation proudly presents its two solutions: x = 0 and x = 1.

These x-values represent the x-coordinates of our elusive intersection points. To unveil the corresponding y-coordinates, we substitute these x-values back into either of the original equations. Choosing the simpler equation, y = 2x - 2, we proceed.

When x = 0, we discover y = 2(0) - 2 = -2. Thus, one solution proudly emerges: (0, -2).

When x = 1, we find y = 2(1) - 2 = 0. And the second solution gracefully unveils itself: (1, 0).

Therefore, with unwavering certainty, we conclude that the solutions to the graphed system of equations y = x^2 + x - 2 and y = 2x - 2 are (0, -2) and (1, 0).

The Power of Visual Representation: Why Graphical Solutions Matter

Graphical solutions transcend mere numerical answers; they offer a profound visual narrative of the relationship between equations and their solutions. They allow us to witness the intersection points, the embodiment of shared values that satisfy the entire system. This visual intuition becomes invaluable when navigating complex equations or systems boasting multiple solutions. The graph serves as a roadmap, guiding us through the landscape of solutions.

Conclusion: Mastering Graphical Solutions

In this comprehensive exploration, we've traversed the realm of solving systems of equations graphically. We've learned that the key lies in identifying the intersection points of the graphs, the points where all equations in the system find common ground. Through the specific example of y = x^2 + x - 2 and y = 2x - 2, we've meticulously demonstrated the process, unveiling the solutions (0, -2) and (1, 0). These points mark the harmonious convergence of the parabola and the line on the coordinate plane.

The ability to solve systems of equations graphically is a cornerstone of mathematical understanding. It provides a visual and intuitive lens through which we can explore the relationships between equations and their solutions. Mastering this skill empowers us to tackle a wide range of mathematical challenges with confidence and clarity.

Unraveling Systems of Equations through Graphical Analysis

Solving systems of equations is a cornerstone of mathematics, and graphical methods offer a powerful and intuitive approach. By visualizing equations as lines and curves on a coordinate plane, we can identify their points of intersection, which represent the solutions that satisfy all equations in the system. This article provides a comprehensive exploration of graphical solutions, focusing on the system y = x^2 + x - 2 and y = 2x - 2 as an illustrative example. We will delve into the nature of each equation, their graphical representations, the significance of intersection points, and the process of accurately determining solutions.

Understanding the Equations: A Foundation for Graphical Analysis

Before we embark on the graphical solution, it's crucial to understand the individual equations and their characteristics. This understanding forms the basis for interpreting their graphical behavior and ultimately identifying solutions.

  • The Quadratic Equation: y = x^2 + x - 2

    The equation y = x^2 + x - 2 is a quadratic equation, recognizable by the presence of the x^2 term. Quadratic equations graph as parabolas, which are U-shaped curves. The coefficients of the equation determine the parabola's shape, direction, and position on the coordinate plane. To visualize this equation, we can plot points by substituting different values of x and calculating the corresponding y values. This process will reveal that the parabola opens upwards due to the positive coefficient of the x^2 term. Furthermore, the parabola intersects the x-axis at two points, representing the roots or zeros of the equation. The vertex, the turning point of the parabola, is also a key characteristic to consider.

  • The Linear Equation: y = 2x - 2

    In contrast to the quadratic equation, y = 2x - 2 is a linear equation, characterized by a constant rate of change. Linear equations graph as straight lines. The coefficient of x, which is 2 in this case, represents the slope of the line, indicating its steepness and direction. The constant term, -2, represents the y-intercept, the point where the line crosses the y-axis. To visualize this equation, we can plot points or use the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The line has a positive slope, meaning it rises from left to right, and it intersects the y-axis at the point (0, -2).

Graphical Representation: Visualizing the Equations

To solve a system of equations graphically, we plot the equations on the same coordinate plane. The resulting graph provides a visual representation of the equations and their relationship. In the case of y = x^2 + x - 2 and y = 2x - 2, we plot the parabola and the line on the same graph. The parabola curves upwards, and the line rises with a positive slope. The points where these two graphs intersect are the solutions to the system of equations.

Intersection Points: The Key to Solutions

The points where the graphs of the equations intersect are the solutions to the system of equations. These intersection points represent the ordered pairs (x, y) that satisfy both equations simultaneously. In other words, when we substitute the x and y values of an intersection point into both equations, the equations hold true. Graphically, these points are where the parabola and the line share a common location on the coordinate plane.

Finding the Solutions: A Step-by-Step Approach

To find the solutions graphically, we can visually inspect the graph and identify the intersection points. However, for a more precise and accurate solution, we can use algebraic methods. One common method is to set the two equations equal to each other. This is based on the principle that at the intersection points, the y values of both equations are the same.

In our example, we set x^2 + x - 2 = 2x - 2. This creates a single equation in terms of x, which we can solve using algebraic techniques.

Subtracting 2x and adding 2 to both sides, we get: x^2 - x = 0.

Factoring out an x, we have: x(x - 1) = 0.

This equation has two solutions: x = 0 and x = 1.

These x values represent the x-coordinates of the intersection points. To find the corresponding y-coordinates, we substitute these x values back into either of the original equations. Choosing the simpler equation, y = 2x - 2, we proceed.

For x = 0, we get y = 2(0) - 2 = -2. So, one solution is (0, -2).

For x = 1, we get y = 2(1) - 2 = 0. So, the other solution is (1, 0).

Therefore, the solutions to the graphed system of equations y = x^2 + x - 2 and y = 2x - 2 are (0, -2) and (1, 0).

The Significance of Graphical Solutions

Graphical solutions provide a valuable visual representation of the relationship between equations and their solutions. They allow us to see the intersection points, which represent the values that satisfy all equations in the system. This visual understanding can be particularly helpful when dealing with more complex equations or systems with multiple solutions. Additionally, graphical solutions can be used to estimate solutions even when algebraic methods are difficult or impossible to apply.

Conclusion: Mastering Graphical Solutions for Systems of Equations

In conclusion, solving systems of equations graphically involves plotting the equations on the same coordinate plane and identifying the points of intersection. These intersection points represent the solutions that satisfy all equations in the system. For the system y = x^2 + x - 2 and y = 2x - 2, we found the solutions to be (0, -2) and (1, 0). Graphical solutions offer a powerful visual tool for understanding and solving systems of equations, providing insights that can be difficult to obtain through algebraic methods alone. Mastering graphical solutions is an essential skill for anyone studying mathematics or related fields.