Problem 37 - Precalculus: Mathematics for Calculus

Question

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

Intersection points: \((1.92, 6.68)\) and \((5.08, 6.08)\).

Step 1: Understand the Equations

The system consists of two equations. The first equation \(\frac{x^{2}}{9}+\frac{y^{2}}{18}=1\) represents an ellipse centered at the origin with a semi-major axis of \(\sqrt{18}\) and a semi-minor axis of \(\sqrt{9}\). The second equation \(y=-x^{2}+6x-2\) is a downward-opening parabola.

Step 2: Sketch the Ellipse

Plot the ellipse described by \(\frac{x^{2}}{9}+\frac{y^{2}}{18}=1\). It is centered at the origin \((0,0)\), with x-intercepts of \(( -3, 0), (3, 0)\) and y-intercepts of \((0, \sqrt{18}), (0, -\sqrt{18})\).

Step 3: Sketch the Parabola

The parabola \(y=-x^{2}+6x-2\) can be sketched by identifying critical points, such as the vertex. Completing the square or using the vertex formula gives the vertex at \((3, 7)\). Additionally, find x-intercepts and y-intercepts if needed. Draw the curve opening downwards accordingly.

Step 4: Determine Intersection Points

Graphically, locate the points where the ellipse and parabola intersect. These points are the solutions of the system of equations. Examine the graph to identify the approximate intersection points visually.

Step 5: Approximate Solutions

Round the identified intersection points to two decimal places. Based on visual inspection, these points appear around \((1.92, 6.68)\) and \((5.08, 6.08)\).

Step 6: Verify Solutions

Plug the approximate coordinates back into both equations to check if they satisfy both equations. If both are approximately true, the points are confirmed as solutions.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share common variables. Solving a system means finding the values of these variables that satisfy all the equations at the same time. In this exercise, we work with the graphical method. This involves plotting each equation on a graph to visually identify where the solutions exist as intersection points.

Using graphs can be a powerful tool to solve systems of equations, especially when the equations are nonlinear. Observing where the graphs intersect gives a tangible sense of the solution. In this scenario, we have one equation representing an ellipse and another representing a parabola. By graphing both, we can see how they overlap in the coordinate plane, pinpointing their common solutions.

Ellipse

An ellipse is a set of points where the sum of the distances from two fixed points, known as foci, is constant. It looks like an elongated circle. Understanding the specific form and position of an ellipse can help greatly when graphing.

For the equation \(\frac{x^{2}}{9} + \frac{y^{2}}{18} = 1\), this represents an ellipse centered at the origin. The semi-major and semi-minor axes give us the dimensions along x and y directions. Here, the semi-major axis length is \(\sqrt{18}\) and semi-minor is \(\sqrt{9}\). This means the ellipse stretches wider vertically than horizontally. Plotting these on the graph gives us the foundational shape upon which to overlay the second curve, the parabola.

Parabola

A parabola is a symmetric curve defined by a quadratic equation, most commonly taking the form \(y = ax^2 + bx + c\). In this system of equations, the parabola is defined by \(y = -x^2 + 6x - 2\). This particular parabola opens downward, indicated by the negative coefficient of \(x^2\).

Knowing key features such as the vertex, axis of symmetry, and intercepts helps to sketch this curve accurately. The vertex, computed to be \((3, 7)\), is the highest point on this downward-opening parabola. The axis of symmetry, a vertical line passing through the vertex, is \(x = 3\). By plotting this on a graph, we can visualize how the parabola and ellipse intersect.

Intersection Points

Intersection points on a graph indicate solutions where two different curves meet. For a system of equations, these are the values that satisfy both equations. Once both the ellipse and parabola are accurately graphed, checking where they overlap is essential.

The solutions are identified by locating these intersection points visually. In this exercise, the approximate intersection points were found at \((1.92, 6.68)\) and \((5.08, 6.08)\). These represent the coordinates on the graph where both the ellipse and the parabola intersect, thereby solving the system. Verifying these solutions involves inserting these coordinates back into the original equations to ensure each is nearly satisfied, ensuring accuracy.