Problem 74 - A Graphical Approach to College Algebra

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

The system has no real solutions due to inconsistency.

Step 1: Analyze the System

The given system of equations is: \( 2x^2 + 3y^2 = 5 \) and \( 4x^2 + 6y^2 = 8 \). Notice that the second equation is exactly twice the first. This indicates the system might have infinitely many solutions or describes the same graph.

Step 2: Simplify the System

We can simplify the second equation by dividing it by 2: \( \frac{4x^2 + 6y^2 = 8}{2} \), resulting in \( 2x^2 + 3y^2 = 4 \). Now compare this with the first equation \( 2x^2 + 3y^2 = 5 \). This simplification shows an inconsistency between the two equations which implies there are no real solutions where both equations hold true simultaneously.

Key Concepts

Analytical Solution

Solving a nonlinear system of equations analytically involves manipulating the given equations to achieve a precise solution. When working with nonlinear systems, you'd primarily deal with polynomials, exponentials, and other combinations that aren't simply linear. The approach is to combine algebraic manipulations
with mathematical insights to derive a solution without relying on numerical approximation. Analytical solutions give us the exact answer, which is crucial for precise results in mathematics and related fields. They help to understand the inherent relationships within the system and often lead to discovering properties
of the equations that are not immediately obvious, such as consistency or the lack thereof. In this exercise, analytical methods involve simplifying equations to assess if they describe the same geometric shape or not.

Real Solutions

Real solutions are specific to systems of equations and indicate the existence of answers that fit within the set of real numbers. For a system of equations, having real solutions means that there's a point (or points) where the equations intersect on the real-number plane. Finding real solutions helps us comprehend whether the quantities described by the equations are tangible or physically realizable.

• Equations have real solutions if they can be satisfied by one or more real numbers.
• If none exist, the system could either be inconsistent or have solutions involving complex numbers.
• Real solutions are vital in real-world applications, such as physics or engineering, where imaginary or complex numbers might not have physical meaning.

In this exercise, analyzing the equations revealed they were inconsistent when looking for real solutions, implying no physical real intersection exists.

Equation Simplification

Simplifying equations is a critical technique used to make complex systems easier to solve. It involves reducing an equation to its simplest form by removing common factors or terms, combining like terms, or performing operations like division or multiplication across the equation. Simplifying helps to:

• Identify relationships, similarities, or inconsistencies between equations.
• Highlight contradictions, which can indicate whether a system has solutions.
• Reduce computational complexity, making it easier to find a solution.

In this case, the simplification of the equation in the step-by-step solution involved dividing the entire second equation, revealing inconsistency. This contradiction showed that these particular equations, despite their simplicity, couldn't have overlapping real solutions as initially hypothesized.