Problem 61 - Algebra and Trigonometry

Question

In Exercises \(61-66,\) find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{l} {x^{2}+y^{2}=1} \\ {x^{2}+9 y^{2}=9} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set of the system is \{(1,0), (-1,0), (0,1), (0,-1)\}.

Step 1: Graph the Equations

The first equation \(x^{2}+y^{2}=1\) represents a circle with a center at the origin and a radius of 1. The second equation \(x^{2}+9 y^{2}=9\) represents an ellipse centered at the origin with a major axis along the x-axis of length 3 and a minor axis along the y-axis of length 1. Graph these equations using a graphing tool.

Step 2: Find the Points of Intersection

The points of intersection of the two graphs are the solutions to the system of equations. Visually, it can be noticed that the graphs intersect at four points: (1,0), (-1,0), (0,1), and (0,-1).

Step 3: Check the Solutions

Substitute the solutions into both the original equations to verify the solutions. Each of the solutions, when plugged into the original equations, validates the equations, confirming the solutions as correct.

Key Concepts

Graphing Systems of Equations

Graphing systems of equations involves plotting multiple equations on the same coordinate plane, with the goal to find where these graphs intersect. This technique is valuable for visual learners who benefit from seeing the graphical representation of equations. Here's how to do it:

• Start by identifying the type of each equation. In this case, we have a circle and an ellipse.
• Plot each equation individually on the graph. Ensure to use correct scales for both the x and y axes.
• Where the graphs intersect, these points are your potential solutions to the system of equations.

Once plotted, it becomes much easier to visually determine the solutions by observing where the graphs cross each other. This not only offers a solution but also a proof of concept through visualization.

Circles and Ellipses

Circles and ellipses are both conic sections, but they appear quite different on a graph.For a circle, the equation is of the form: \[ x^2 + y^2 = r^2 \]where

• the center is at the origin (0,0), and
• the radius is \( r \).

An ellipse has the equation: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where:

• \( a \) is the semi-major axis,
• \( b \) is the semi-minor axis, and
• the center is also at the origin if it's not shifted.

In this exercise, when you graph both a circle and an ellipse, their unique properties will guide your interpretation of the graph, showing distinct points of intersection.

Points of Intersection

Finding points of intersection between graphs is like looking for solutions common to both equations. It means finding coordinates where both functions have the same values of \( x \) and \( y \).Here's a practical approach:

• Once both graphs are plotted, observe where they cross each other.
• These crossing points will give you an immediate list of potential solutions.
• Verify by substituting these points into the original equations to check their validity.

In the given exercise, the circle and the ellipse intersect at four points: (1,0), (-1,0), (0,1), and (0,-1). This tells us these are possible values where both equations are satisfied at the same time.

Algebra and Trigonometry

Algebra and trigonometry are fundamental to understanding systems of equations involving circles and ellipses.In algebra, systems of equations are tackled by methods like substitution or elimination. However, graphing provides a clear visual representation, which can be more intuitive at times.From a trigonometric perspective, circles relate to trigonometric functions because of their round shape and angles. For instance, the equation of a circle \( x^2 + y^2 = 1 \) can be related to the unit circle, with trigonometric identities like \( \sin^2\theta + \cos^2\theta = 1 \).By understanding these algebraic and trigonometric elements, solving and graphing systems becomes manageable, offering insights into both the rules and applications of mathematics.