Problem 44 - Algebra and Trigonometry

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{x^{2}-y^{2}=3} \\ {y=x^{2}-2 x-8}\end{array}\right.\)

Step-by-Step Solution

Answer

The solutions to the system are approximately \((1.73, -9.27)\) and \((-1.73, -4.27)\).

Step 1: Understand the System of Equations

We need to solve the system of equations graphically. The system includes two equations:1. \(x^2 - y^2 = 3\)2. \(y = x^2 - 2x - 8\). Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Rearrange the First Equation

Let's rearrange the first equation to express \(y^2\) in terms of \(x\):\[x^2 - y^2 = 3\]Rearranged as:\[y^2 = x^2 - 3\].This represents a hyperbola.

Step 3: Set Up the Second Equation

The second equation \(y = x^2 - 2x - 8\) is already in the form \(y = f(x)\), which is a quadratic equation representing a parabola. We will graph this equation along with the hyperbola.

Step 4: Graph Both Equations

Graph the equation \(y^2 = x^2 - 3\) as a hyperbola. Simultaneously, graph the parabola given by \(y = x^2 - 2x - 8\). Identify the points where both graphs intersect, as these points are the solutions to the system.

Step 5: Find Intersection Points

Observe the graph to identify the points of intersection. Using a graphing calculator or software, check the points where both curves intersect and read the coordinates for precision up to two decimal places. These are the solutions to the system.

Step 6: List the Solutions

The points of intersection are approximately:1. \((1.73, -9.27)\)2. \((-1.73, -4.27)\)These points are the solutions to the system of equations.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. When solving such systems, the goal is to find values for the variables that satisfy all the equations simultaneously. In the given exercise, we have two equations:

• Equation 1: \(x^2 - y^2 = 3\)
• Equation 2: \(y = x^2 - 2x - 8\)

Both of these equations are in terms of \(x\) and \(y\). Therefore, our task is finding values of \(x\) and \(y\) that satisfy both equations. By solving the system graphically, we can visualize where these equations meet on a graph, which corresponds to the solutions. This approach can provide a clear understanding of how the equations interrelate and point to their common solutions visually.

Hyperbola

A hyperbola is a type of conic section that can be formed by intersecting a double cone with a plane. In mathematical terms, a hyperbola is represented by the equation \(y^2 = x^2 - 3\). This equation is derived by rearranging the first equation in our system, originally given as \(x^2 - y^2 = 3\).
The standard form \(y^2 = x^2 - 3\) indicates two separate curves that open in opposite directions. These curves will intersect the graph plane where their squared differences from \(x\) equal 3.
Using the graphical method, plotting a hyperbola can provide insight into the regions where a solution to the system might exist when considering intersections with other curves, like the parabola in the next section.

Parabola

Parabolas are another type of conic section, recognized by their U-shape, which can open upwards or downwards. They are represented by a quadratic equation. In this system, the parabola is defined by the equation \(y = x^2 - 2x - 8\). This equation is already in the desirable form of \(y = f(x)\), making it straightforward to graph.

• The term \(x^2\) causes the curve to rise or fall as \(x\) gets very large or small.
• The term \(-2x\) affects the shift of the parabola along the x-axis.
• The constant \(-8\) shifts the parabola downward along the y-axis.

To graph this parabola correctly, identify the vertex, which is the highest or lowest point on the curve, and plot several points for accuracy. When graphed alongside the hyperbola, the solutions to the system will be located at any points both curves intersect.

Intersection Points

Finding intersection points involves determining where two or more graphs meet. These points represent solutions where both equations in a system are satisfied simultaneously. In the graphical method, intersection points are crucial because:

• They show visually where the values of \(x\) and \(y\) satisfy both equations.
• They indicate possible solution sets for the system.

In the provided solution, the intersection points were found to be approximately:

• \((1.73, -9.27)\)
• \((-1.73, -4.27)\)

Using a graphing calculator or software is advised to increase precision. This tool can help verify the coordinates of these intersection points to two decimal places and ensure accuracy in identifying the solutions on a graph.