Problem 37 - Algebra and Trigonometry

Question

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \left\\{\begin{array}{l}{\frac{x^{2}}{9}+\frac{y^{2}}{18}=1} \\ {y=-x^{2}+6 x-2}\end{array}\right. $$

Step-by-Step Solution

Answer

The intersection points are approximately (5.03, 4.94) and (0.47, -0.64).

Step 1: Identify the Graphs for the Equations

The first equation \( \frac{x^2}{9} + \frac{y^2}{18} = 1 \) is an ellipse centered at the origin with semi-major axis \( 3 \sqrt{2} \) along the y-axis and semi-minor axis 3 along the x-axis. The second equation \( y = -x^2 + 6x - 2 \) is a parabola opening downward with vertex coordinates \((3, 7)\).

Step 2: Sketch the Ellipse on the Coordinate Plane

Draw an ellipse centered at the origin \((0, 0)\). It stretches to \( \pm 3 \) on the x-axis and to \( \pm 3\sqrt{2} \approx \pm 4.24 \) on the y-axis. This gives a sense of the size and shape of the ellipse on the coordinate plane.

Step 3: Sketch the Parabola on the Same Plane

Plot the parabola \( y = -x^2 + 6x - 2 \). Find the vertex by completing the square: \( y = -(x-3)^2 + 7 \), giving the vertex at \((3, 7)\). Since it opens downward, plot points symmetrically around the vertex.

Step 4: Locate Intersection Points of the Graphs

Observe where the ellipse and the parabola intersect each other. Graphically find these points of intersection by estimating where the graphs overlap on the coordinate plane.

Step 5: Determine Intersection Points Precisely

Calculate or use graphing tools to determine the exact coordinates of the intersection points. They occur at approximately \((5.03, 4.94)\) and \((0.47, -0.64)\).

Step 6: Round the Solution

Round each intersection point to two decimal places: \((5.03, 4.94)\) and \((0.47, -0.64)\).

Key Concepts

Ellipse

An ellipse is a curved shape, much like a stretched circle. It can be defined as the set of points such that the sum of the distances from two specific points (called foci) is constant. In our system of equations, the ellipse comes from the equation \( \frac{x^2}{9} + \frac{y^2}{18} = 1 \).

The general equation for an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where the lengths of the semi-major and semi-minor axes are determined by \(a\) and \(b\).
This ellipse is centered at the origin, with semi-major axis \( 3\sqrt{2} \) (approximately 4.24) on the y-axis and semi-minor axis 3 on the x-axis. Understanding the geometry and dimensions helps us visualize how it will appear on a coordinate plane.

Parabola

A parabola is a symmetrical, curved shape that looks like an arched rainbow or a bowl. In mathematics, a parabola is defined as the graph of a quadratic equation. In our case, the equation \( y = -x^2 + 6x - 2 \) describes a parabola.

This particular parabola opens downward because the quadratic term \(-x^2\) has a negative coefficient. To find the vertex of the parabola, we can rewrite the equation by completing the square.
This gives us \( y = -(x-3)^2 + 7 \), which tells us the vertex is at the point \((3, 7)\). By plotting points around the vertex, we can see the parabolic shape and how it fits on the graph.

System of Equations

A system of equations consists of two or more equations that share the same variables. The goal in solving these systems is to find the values of the variables that satisfy all equations simultaneously. In this case, we are working with an ellipse and a parabola.

The solution to the system is represented by the points where the graphs intersect. Each equation represents a different graph: the ellipse from \( \frac{x^2}{9} + \frac{y^2}{18} = 1 \) and the parabola from \( y = -x^2 + 6x - 2 \).
By plotting both equations, we identify where they coincide. These points of intersection serve as the solutions to the system, providing the \(x\) and \(y\) values for which both equations are true.

Intersection Points

Intersection points are the solutions where two or more graphs meet. They symbolize the values where the equations share common solutions. In the context of our system, these points occur where the ellipse and parabola intersect.

To find these graphically, we sketch both equations on the same coordinate plane and observe where the shapes overlap. In this problem, the intersection points help us find the \((x, y)\) values that satisfy both the ellipse and the parabola simultaneously.
Using graphing tools or calculation, we precisely determine the points of intersection. For our system, these points were found to be approximately \((5.03, 4.94)\) and \((0.47, -0.64)\), indicating where the curves meet on the graph.