Graphical Solution Of System Of Equations X² + Y² = 4 And X - Y = 1

In the realm of mathematics, solving systems of equations often involves finding the points where the graphs of the equations intersect. This article delves into the graphical representation of the solution for the system of equations: x² + y² = 4 and x - y = 1. We will explore each equation individually, understand their graphical forms, and then determine the points of intersection, which represent the solution to the system. This process not only enhances our understanding of algebraic solutions but also provides a visual interpretation of the results. By understanding how to solve these systems, we can apply these skills to more complex problems in mathematics and real-world applications.

Understanding the Equations

Equation 1: x² + y² = 4

The first equation, x² + y² = 4, is a fundamental representation of a circle in the Cartesian plane. In this equation, we can immediately recognize the standard form of a circle's equation: (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius. In our case, the equation simplifies to x² + y² = 4, indicating that the center of the circle is at the origin (0, 0) and the radius is the square root of 4, which is 2. Therefore, the graph of this equation is a circle centered at the origin with a radius of 2 units.

To visualize this circle, imagine a compass placed at the origin of the coordinate plane. With the compass extended to a length of 2 units, trace a full rotation. The resulting shape is the circle represented by the equation x² + y² = 4. Every point (x, y) on this circle satisfies the equation, meaning that when the x and y coordinates are substituted into the equation, the equality holds true. For example, the points (2, 0), (0, 2), (-2, 0), and (0, -2) all lie on this circle, as do infinitely many other points. Understanding the circle's equation is crucial for grasping the graphical solution of the system because it sets the stage for identifying the points where the circle intersects with the other equation, which we will explore next.

Equation 2: x - y = 1

The second equation, x - y = 1, represents a straight line in the Cartesian plane. To better understand this linear equation, we can rearrange it into the slope-intercept form, which is y = mx + b, where m is the slope of the line and b is the y-intercept. Rearranging x - y = 1, we get y = x - 1. From this form, we can easily identify that the slope (m) is 1 and the y-intercept (b) is -1. This means that the line rises one unit vertically for every one unit it moves horizontally, and it crosses the y-axis at the point (0, -1).

To graph this line, we can start by plotting the y-intercept at (0, -1). Since the slope is 1, we can move one unit to the right and one unit up from the y-intercept to find another point on the line, such as (1, 0). Connecting these two points with a straight line gives us the graphical representation of the equation x - y = 1. Every point (x, y) on this line satisfies the equation, meaning that when the x and y coordinates are substituted into the equation, the equality holds true. Understanding the line's equation is essential for finding the graphical solution of the system, as we need to determine where this line intersects with the circle represented by the first equation. The points of intersection will give us the solutions to the system of equations.

Graphical Representation and Intersection

To graphically solve the system of equations, x² + y² = 4 and x - y = 1, we need to visualize the graphs of both equations on the same coordinate plane and identify their points of intersection. The first equation, x² + y² = 4, represents a circle centered at the origin (0, 0) with a radius of 2. The second equation, x - y = 1, represents a straight line with a slope of 1 and a y-intercept of -1. When these two graphs are plotted on the same plane, they will intersect at one or more points, or they might not intersect at all, depending on the specific equations.

The circle, with its center at the origin and a radius of 2, spans across the coordinate plane, touching the x-axis at (-2, 0) and (2, 0) and the y-axis at (0, -2) and (0, 2). The line, with its slope of 1 and y-intercept of -1, rises diagonally across the plane. Visually, it becomes clear that the line will intersect the circle at two distinct points. These points of intersection are the solutions to the system of equations, as they satisfy both equations simultaneously. To find these points precisely, we would typically solve the system algebraically, either by substitution or elimination. However, the graphical representation provides a clear visual confirmation of the number of solutions and their approximate locations. By plotting the graphs accurately, we can estimate the coordinates of the intersection points, which can then be verified through algebraic methods.

Determining the Solution

The points where the circle and the line intersect graphically represent the solutions to the system of equations. These points satisfy both the equation of the circle (x² + y² = 4) and the equation of the line (x - y = 1). To find these points precisely, we can solve the system of equations algebraically. There are two primary methods for solving such systems: substitution and elimination. In this case, the substitution method is particularly effective.

First, we can rearrange the linear equation (x - y = 1) to express one variable in terms of the other. Let's solve for x: x = y + 1. Now, we can substitute this expression for x into the equation of the circle: (y + 1)² + y² = 4. Expanding and simplifying this equation, we get: y² + 2y + 1 + y² = 4, which further simplifies to 2y² + 2y - 3 = 0. This is a quadratic equation in terms of y, which we can solve using the quadratic formula: y = [-b ± √(b² - 4ac)] / (2a). In our case, a = 2, b = 2, and c = -3. Plugging these values into the quadratic formula, we get: y = [-2 ± √(2² - 4 * 2 * -3)] / (2 * 2). Simplifying further, we have: y = [-2 ± √(4 + 24)] / 4, which gives us: y = [-2 ± √28] / 4. This yields two possible values for y: y = (-2 + √28) / 4 and y = (-2 - √28) / 4. These values can be further simplified to y = (-1 + √7) / 2 and y = (-1 - √7) / 2.

Now that we have the y-coordinates of the intersection points, we can substitute these values back into the equation x = y + 1 to find the corresponding x-coordinates. For y = (-1 + √7) / 2, we have: x = [(-1 + √7) / 2] + 1, which simplifies to x = (1 + √7) / 2. For y = (-1 - √7) / 2, we have: x = [(-1 - √7) / 2] + 1, which simplifies to x = (1 - √7) / 2. Therefore, the two points of intersection, which represent the solutions to the system of equations, are ((1 + √7) / 2, (-1 + √7) / 2) and ((1 - √7) / 2, (-1 - √7) / 2). These points can be plotted on the graph to verify their location and confirm the graphical solution.

Conclusion

In conclusion, the graphical representation of the system of equations x² + y² = 4 and x - y = 1 involves plotting a circle and a line on the same coordinate plane. The circle, centered at the origin with a radius of 2, and the line, with a slope of 1 and a y-intercept of -1, intersect at two distinct points. These points of intersection graphically represent the solutions to the system of equations. By solving the system algebraically, using the substitution method, we found the exact coordinates of these points to be ((1 + √7) / 2, (-1 + √7) / 2) and ((1 - √7) / 2, (-1 - √7) / 2).

The graphical method provides a visual confirmation of the number of solutions and their approximate locations, while the algebraic method allows us to determine the precise coordinates. Understanding both methods is crucial for solving systems of equations effectively. The intersection points not only satisfy both equations but also offer valuable insights into the relationships between the circle and the line. This process of solving systems of equations graphically and algebraically is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, and computer science. The ability to interpret and solve these systems enhances our problem-solving capabilities and our understanding of mathematical concepts.