Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &2 x^{2}+2 y^{2}=20\\\ &3 x^{2}+3 y^{2}=30 \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are all points \((x, y)\) on the circle \(x^2 + y^2 = 10\).

Step 1: Analyze the equations

First, let's examine both equations. The first equation is \(2x^2 + 2y^2 = 20\) and the second is \(3x^2 + 3y^2 = 30\). Notice that both equations resemble the equation of a circle, but the constant term on the right side differs.

Step 2: Simplify the equations

Divide the first equation by 2: \(x^2 + y^2 = 10\). Divide the second equation by 3: \(x^2 + y^2 = 10\). Both equations simplify to the same equation.

Step 3: Identify the solutions from the simplified equation

Since both equations simplify to \(x^2 + y^2 = 10\), we recognize this as the equation of a circle centered at the origin with a radius \(\sqrt{10}\). The solutions are all points \((x, y)\) that lie on this circle.

Key Concepts

Analytical Solutions

An analytical solution refers to finding the exact answers to equations or mathematical problems using algebraic manipulations or formulas. Unlike numerical solutions, which approximate solutions using algorithms and iterations, analytical solutions give precise results. In this particular exercise, both equations are examined, and algebraic techniques are used to simplify them to a more manageable form. This allows us to derive exact and general solutions for the system without resorting to approximations.

When dealing with systems of equations, an analytical approach often involves:

• Simplifying each equation.
• Utilizing substitution or elimination methods to solve the system.
• Recognizing patterns that may align with known algebraic identities or geometric shapes, as seen with circles here.

In this scenario, after dividing and simplifying the equations, both reduce to a circle equation, leading to straightforward identification of solutions.

Equations of a Circle

The equation of a circle in standard form is given by \[x^2 + y^2 = r^2\] where - the center of the circle is at the origin (0, 0)- and the radius is \(r\).

Understanding this concept helps in geometrically interpreting equations that appear in algebra problems. For the given problem, both equations simplify to \(x^2 + y^2 = 10\), indicating a circle with a center at the origin and a radius of \(\sqrt{10}\). All real solutions of this equation correspond to all \((x, y)\) points lying on the circumference of this circle.

Visualizing such equations is crucial when solving systems involving geometrical figures. Here, recognizing the pattern of a circle from the simplified equation not only makes solving the problem more intuitive, but it also provides geometric insight into the distribution of solutions on the coordinate plane.

Solving Systems of Equations

Solving systems of nonlinear equations involves finding values of variables that satisfy all equations simultaneously. In this task, both equations represented circles, which is a typical nonlinear system scenario. To solve them, follow these general steps:

• Identify the type of equations you are dealing with (e.g., linear, quadratic, or in this case, circular).
• Simplify the equations if possible to find a common form or pattern.
• Use substitution or elimination to reduce the system to fewer variables if needed.
• Calculate or determine the points that satisfy all original equations.

For this problem, the simplifying revealed a single equation, \(x^2 + y^2 = 10\), from both. This simplification implied that all solutions to the system are valid at any point on the circle defined by this equation. Understanding the nature of these equations allows us to graphically interpret and solve the problem.