Find The Solution(s) Of The System Of Equations $y = X^2 - X + 1$ And $y = X$ Using Graphing.

by ADMIN 94 views

In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields. This article delves into the specific problem of finding the solutions to the system of equations y = x² - x + 1 and y = x. We will explore how to determine the solution set by employing the graphical method, providing a visual understanding of the intersection points that represent the solutions. Mastering this technique empowers you to tackle more complex mathematical challenges and gain a deeper appreciation for the interplay between algebraic expressions and their graphical representations.

H2: Understanding the Equations

Before diving into the graphical solution, let's take a closer look at the equations themselves. We have two equations:

  1. y = x² - x + 1: This is a quadratic equation, which, when graphed, represents a parabola. The shape of the parabola depends on the coefficient of the x² term (which is 1 in this case, indicating an upward-opening parabola) and the other terms that influence its position and width.
  2. y = x: This is a linear equation, and its graph is a straight line that passes through the origin with a slope of 1. It represents a direct proportional relationship between x and y.

The solutions to this system of equations are the points where the graphs of these two equations intersect. At these intersection points, the x and y values satisfy both equations simultaneously. To find these points, we will employ the graphical method, which involves plotting both equations on the same coordinate plane and identifying their points of intersection.

H3: Plotting the Parabola (y = x² - x + 1)

To accurately plot the parabola, we need to determine several key points. These include the vertex (the turning point of the parabola), the y-intercept (where the parabola crosses the y-axis), and a few other points to get a sense of the curve's shape. Understanding these features allows for a precise representation of the quadratic equation on the coordinate plane.

  • Vertex: The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a and b are the coefficients of the x² and x terms, respectively. In this case, a = 1 and b = -1, so the x-coordinate of the vertex is x = -(-1) / (2 * 1) = 1/2. To find the y-coordinate, we substitute this x value back into the equation: y = (1/2)² - (1/2) + 1 = 1/4 - 1/2 + 1 = 3/4. Therefore, the vertex is at the point (1/2, 3/4).
  • Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. Substituting x = 0 into the equation, we get y = 0² - 0 + 1 = 1. So the y-intercept is at the point (0, 1).
  • Additional Points: To get a better sense of the parabola's shape, we can choose a few additional x values and calculate the corresponding y values. For example:
    • If x = 1, then y = 1² - 1 + 1 = 1. This gives us the point (1, 1).
    • If x = -1, then y = (-1)² - (-1) + 1 = 1 + 1 + 1 = 3. This gives us the point (-1, 3).

By plotting these points and connecting them with a smooth curve, we obtain the graph of the parabola y = x² - x + 1. The careful calculation and plotting of these points are crucial for accurately visualizing the quadratic equation and its relationship with the linear equation.

H3: Plotting the Line (y = x)

The linear equation y = x is straightforward to plot. It's a straight line that passes through the origin (0, 0) and has a slope of 1. This means that for every unit increase in x, y also increases by one unit. We can identify a few points on this line:

  • (0, 0)
  • (1, 1)
  • (2, 2)
  • (-1, -1)

Plotting these points and drawing a straight line through them gives us the graph of y = x. The simplicity of this line makes it a valuable reference point when analyzing its intersection with the parabola.

H2: Identifying the Intersection Points

Now that we have plotted both the parabola and the line on the same coordinate plane, the solutions to the system of equations are the points where the two graphs intersect. By visually inspecting the graph, we can identify the intersection point(s).

In this case, the parabola and the line intersect at only one point: (1, 1). This point lies on both the parabola and the line, meaning that the x and y values at this point satisfy both equations simultaneously. The accurate identification of this intersection point is the key to solving the system of equations graphically.

H2: Verifying the Solution

To ensure that (1, 1) is indeed a solution, we can substitute these values into both equations:

  • For y = x² - x + 1: 1 = (1)² - 1 + 1 => 1 = 1 - 1 + 1 => 1 = 1 (True)
  • For y = x: 1 = 1 (True)

Since the point (1, 1) satisfies both equations, it is a valid solution to the system. This verification step is essential to confirm the accuracy of the graphical solution and to solidify our understanding of the algebraic-geometric connection.

H2: Conclusion

Therefore, the solution set for the system of equations y = x² - x + 1 and y = x is {(1, 1)}. This means that the only point where the graphs of these two equations intersect is at the coordinates (1, 1). By graphing the equations and visually identifying their intersection, we have successfully determined the solution to this system. This graphical approach not only provides the answer but also enhances our understanding of how different types of equations relate to each other geometrically. Mastering these skills opens up a pathway to solving more complex mathematical problems and appreciating the visual nature of algebra.

The correct answer is A. (1,1)