Problem 65 - 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. $$ \left\\{\begin{array}{c} {4 x^{2}+y^{2}=4} \\ {2 x-y=2} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set for the system of equations can be found by graphing the equations and identifying the points of intersection. It's important to check the solution in both equations to confirm the results.

Step 1: Analyze and Rewrite the Equations

The first equation \(4 x^{2}+y^{2}=4\) is an equation of an ellipse with the center at the origin. Divide both sides of the equation by 4 to put it in standard form: \(x^{2}+\frac{y^{2}}{4}=1\).\nThe second equation \(2 x-y=2\) is a linear equation. Rewrite the equation to express y in terms of x: \(y = 2x - 2\).

Step 2: Graph the Equations

After the equations have been put in the correct form, plot both the ellipse and the line on the same graph. This can be done by hand or with a graphing calculator or software. Using a computer is recommended for accuracy.

Step 3: Find Intersection Points

Look for the points where the line intersects the ellipse. These points are the solutions to the system of equations. Depending on the specific line and ellipse, there may be two, one, or zero solutions.

Step 4: Check the Solutions

Substitute the x, y coordinates of the intersection points into both original equations to verify that they satisfy the equations. This ensures the solutions are correct.

Key Concepts

Graphing Method

The graphing method is a valuable technique used to find solutions to a system of equations. This involves plotting each equation on the same set of axes and identifying where the graphs intersect. This intersection point (or points) gives the solution(s) to the system.

In this exercise:

• We have two equations, one representing an ellipse and the other a line.
• By graphing these equations, we look for intersection points, which indicate the x and y values satisfying both equations.

Graphing can be done manually for a basic understanding or aided by graphing tools for precision. This method is intuitive as it visually demonstrates the relationship between the equations.

Ellipse

An ellipse is a type of curve on a plane, surrounding two focal points, with the property that the sum of the distances to the two focal points is constant for every point on the curve.

In the equation:

• The given form is \[4x^2 + y^2 = 4\]
• To convert it to the standard form of an ellipse, divide by 4:\[x^2 + \frac{y^2}{4} = 1\]

This standard form equation \[x^2/a^2 + y^2/b^2 = 1\]represents an ellipse, where 'a' and 'b' are the semi-major and semi-minor axes lengths. Here, the ellipse is centered at the origin (0,0) with 'a' being 1 and 'b' being 2. The equation reveals information about the ellipse's size and orientation.

Intersection

The intersection of a line and an ellipse is found where both the line's and ellipse's graphs cross each other. This crossing point(s) provides the solution(s) of the system.

In our example, we analyze:

• The line: \[y = 2x - 2\]
• The ellipse given by \[x^2 + \frac{y^2}{4} = 1\]

After graphing, the points of intersection denote solutions that satisfy both equations simultaneously. Depending on their position, lines can intersect with ellipses zero, one, or two times, presenting possible multiple solutions.

Linear Equation

A linear equation represents a straight line in two dimensions and can typically be written in the form \[y = mx + b\], where 'm' is the slope, and 'b' is the y-intercept.

In this exercise, we begin with the equation:

• \[2x - y = 2\]
• Rearrange to find y in terms of x, getting \[y = 2x - 2\]

This format makes it easier to graph, showing how 'y' changes with 'x'. The slope (m=2) indicates the steepness and direction of the line, and the y-intercept (-2) shows where the line crosses the y-axis. These characteristics help determine how the line will behave when included in the graph with the ellipse.