Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &2 x^{2}+3 y^{2}=5\\\ &4 x^{2}+6 y^{2}=8 \end{aligned}$$

Step-by-Step Solution

Answer

There are no real solutions as the system is contradictory.

Step 1: Observe the System of Equations

We are given the system:\[ \begin{align*} 2x^2 + 3y^2 &= 5 \ 4x^2 + 6y^2 &= 8 \end{align*} \] The second equation is exactly double the first equation.

Step 2: Simplify the System

The second equation, \(4x^2 + 6y^2 = 8\), can be simplified by dividing every term by 2, resulting in \(2x^2 + 3y^2 = 4\).

Step 3: Analyze Resulting Equations

We now have two equations: \(2x^2 + 3y^2 = 5\) and \(2x^2 + 3y^2 = 4\). However, our simplification reveals a contradiction because the same expression cannot equal two different values.

Step 4: Conclude on Solutions

Since the simplified equations \(2x^2 + 3y^2 = 5\) and \(2x^2 + 3y^2 = 4\) are contradictory, there are no real solutions that satisfy both equations simultaneously.

Key Concepts

Analytical Solving

When solving systems of equations, especially nonlinear ones, we often rely on analytical methods. This means using algebraic techniques to find exact solutions. In the provided exercise, we are asked to solve a nonlinear system of equations, a pair of quadratic equations in this case. To solve such a system analytically, we aim to manipulate the expressions using logic and algebraic steps rather than approximations or numerical methods.
A key strategy here involves simplifying and comparing the equations. Initially, we notice the second equation is simply a multiple of the first, indicating a redundancy that must be investigated. By analytically comparing and simplifying, you might find contradictions or redundancies that lead to a better understanding of the potential solutions (or lack thereof).
This approach ensures that every step is clear and grounded in algebra, allowing us to identify exact solutions or recognize when none exist.

Real Solutions

Real solutions are the solutions to an equation that are real numbers, as opposed to complex or imaginary numbers. When asked to find real solutions for a system of equations, we are interested only in those solutions that can be plotted on a typical Cartesian plane and have practical applications.
In the context of the given system of equations, finding real solutions means determining values of \(x\) and \(y\) that satisfy both equations simultaneously without reaching contradictions. After simplification, if a contradiction appears, it often indicates that no real solutions exist within the realm of real numbers.
In our exercise, simplifying both equations led to a situation where they seemed to express two different things simultaneously for the same expressions, highlighting that there can be no satisfactory solutions. Therefore, always check for contradictions when seeking real solutions.

Contradiction in Equations

Solving a system sometimes unveils a contradiction, as it did in this exercise. A contradiction occurs when the simplified form of equations suggests two mutually exclusive truths. In our system, upon dividing the second equation by 2, we noticed that both simplified equations were identical in form but with different constants: \(2x^2 + 3y^2 = 5\) and \(2x^2 + 3y^2 = 4\).
Identifying these contradictions is crucial as they indicate scenarios where no shared solution exists, leading to the conclusion that no real values for \(x\) and \(y\) can satisfy both conditions at the same time.
It is essential to clearly document and understand such contradictions to either re-evaluate assumptions and manipulations or confirm that the system is unsolvable in its given form. Always look for these discrepancies, as they are key in solving nonlinear systems of equations analytically.