Problem 48 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(45-48,\) use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places. $$\left\\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {2 x^{2}-y=2}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations are the points of intersection, accurately determined to two decimal places, as displayed on the graphing utility.

Step 1: Set up the graph

Input both equations into a graphing utility separately to generate two different graphs.

Step 2: Identify intersection points

Examine the graph closely and identify the points of intersection. These are the solutions to the system of equations.

Step 3: Precisely determine intersection points

Use the 'intersection' or a similar function on the graphing utility to find the exact coordinates of the intersection points. Ensure they are rounded to two decimal places as per the problem requirement

Key Concepts

System of Equations

A system of equations consists of two or more equations that share common variables. Systems can be linear, quadratic, or any combination of different types of equations. The main goal is to find values for the variables that satisfy all the equations simultaneously. In our exercise, the system includes a quadratic equation, \(x^2 + y^2 = 4\), which represents a circle, and another quadratic equation, \(2x^2 - y = 2\). The solution to the system is found by determining where these equations intersect. This intersection can provide either one solution, multiple solutions, or no solution at all, depending on the nature of the equations involved.

To solve the system of equations, we often use various methods: algebraic substitution, elimination, or graphical methods. In this exercise, the graphical approach is utilized to visually determine the solutions. This approach is especially helpful when dealing with non-linear equations like these, providing a visual representation that highlights any potential solutions as points where the graphs overlap.

Intersection Points

Intersection points are crucial in understanding systems of equations graphically. They represent the exact locations where two or more graphs cross or meet. Each intersection point corresponds to a set of values that satisfy all equations in the system. In a simple scenario, you might find that two circles, lines, or parabolas intersect at zero, one, or more points. In our problem, we are tasked with using a graphing utility to determine these intersection points.

When using a graphing utility, it is important to accurately identify these points, as they are the solutions to our system. Most tools provide functions to automate this process, such as an 'intersection' feature. This feature helps in pinpointing the coordinates where the graphs meet. Accuracy is key; therefore, rounding these intersection coordinates to two decimal places ensures a precise answer.

• intersection points are solutions
• graphing utilities simplify finding these points
• accuracy in locating and rounding is crucial

Quadratic Equations

Quadratic equations are a type of polynomial equation where the variable is squared (\(ax^2 + bx + c = 0\)). They can represent various types of conic sections when graphically plotted. In our exercise, the equations given include quadratic terms, defining shapes such as circles or parabolas.

The first equation, \(x^2 + y^2 = 4\), represents a circle centered at the origin with a radius of 2. The second equation, \(2x^2 - y = 2\), is a parabola that opens upwards. These conic sections create interesting points of intersection that define the solutions to the system. The circle and parabola can intersect in different ways, including no intersection, a single point, or multiple points.

Understanding the nature of quadratic equations is essential for solving the system graphically, as it helps predict the shape and position of the graphs. With a graphing utility, observing these graphs can provide insight into where the intersections occur, allowing for the verification of solutions.