Problem 41 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{l}{3 x-2 y=0} \\ {x^{2}-y^{2}=4}\end{array}\right.$$

Step-by-Step Solution

Answer

The graphical solution to this system of equations would require solving for the points of intersection of the line graph and the circle graph on a set of axes.

Step 1: Rearrange Equations

Firstly, let's rearrange each equation. For the first equation, solve for y: \(y = 1.5x\). For the second equation: \(y = \sqrt{x^{2} - 4}\) and \(y = -\sqrt{x^{2} - 4}\). This second equation represents an upward-facing circle and a downward-facing circle.

Step 2: Plot the Graphs

Next, plot the three equations on the same graph. For the line y = 1.5x: This line will start from the origin and with a positive slope of 1.5 will extend in the first and third quadrants.For the circles y = \sqrt{x^{2} - 4} and y = -\sqrt{x^{2} - 4}: The center of the circles will be at (0,0) with a radius of 2 units, one circle will be in the first and second quadrant, while the other will be in the third and fourth quadrant.

Step 3: Identifying the Solution

The solutions of the system of equations are points where the graph of y = 1.5x intersects the graphs of y = \sqrt{x^{2} - 4} and y = -\sqrt{x^{2} - 4}. These points will be the x-values where x = 1.5y equals x = \sqrt{x^{2} - 4} and x = 1.5y equals x = -\sqrt{x^{2} - 4}

Key Concepts

Understanding Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. In our exercise, we have the equation \(3x - 2y = 0\). To solve it graphically, we rearrange it into the slope-intercept form \(y = mx + b\). Here, we solved it to find \(y = 1.5x\). The equation defines a straight line with:

• a slope of 1.5
• a y-intercept at the origin (0,0)

Graphically, each equation forms a line on the Cartesian plane. By finding where these lines intersect, we identify the solution to the system of linear equations. In cases where more than one line is involved, intersecting lines suggest that there is at least one solution that satisfies all equations simultaneously.

Introduction to Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically expressed in the form \(ax^2 + bx + c = 0\). In this exercise, our quadratic part is expressed with circles. The specific equation \(x^2 - y^2 = 4\) can be rearranged into a circular format:

• \(y = \sqrt{x^2 - 4}\)
• \(y = -\sqrt{x^2 - 4}\)

These forms represent the top and bottom halves of a circle. Although this exercise manipulates it slightly differently, understanding that it essentially creates two possible values for \(y\) at each \(x\) allows us to graph this equation correctly. Circles in coordinate geometry specifically hold the property that each point's distance from the origin or any center is constant.

Exploring Coordinate Geometry

Coordinate geometry, or analytic geometry, involves the study of geometry using a coordinate system. In this plane, we use equations to define geometric shapes, and we visualize these with x and y coordinates. In our exercise, we have:

• A line: \(y = 1.5x\)
• Curves from \(x^2 - y^2 = 4\), rearranged as circles

It requires plotting these graphs to see their relationship visually. Typically, this involves plotting several points for each equation and then connecting these to form the shapes. The line will appear diagonal across the plane, while the curves will manifest as distinct parts of circles. Coordinate geometry gives us a precise framework for understanding and finding points of interest like intersection points.

Determining Intersection Points

Intersection points occur at the x and y values where two or more graphs meet. In the context of our exercise, we need to find where the line \(y = 1.5x\) intersects the curves given by our modified quadratic equation. Intersections are significant because:

• They represent solutions that satisfy all equations simultaneously
• Graphically, it's where the plots visibly meet

To find these, solve both equations simultaneously either analytically or by observing the graph. On plotting, you should notice two intersections. Dispatching these points gives us a graphical solution, highlighting the effectiveness of visual methods in evaluating solutions.