Solving the Simultaneous Equations: $xy^2 = 4$ and $xy = 3$

Introduction

Key Concepts

Understanding Simultaneous Equations

Simultaneous equations involve finding the values of variables that satisfy multiple equations simultaneously. These equations can be linear or nonlinear. The given system comprises a quadratic equation and a linear equation, making it a nonlinear system. Solving such systems often requires substitution or elimination methods tailored to handle the nonlinear nature.

Setting Up the Equations

Consider the system: $$ \begin{align} xy^2 &= 4 \quad \text{(1)} \\ xy &= 3 \quad \text{(2)} \end{align} $$ Equation (2) is linear in $x$ and $y$, while Equation (1) is quadratic due to the $y^2$ term. The goal is to find the values of $x$ and $y$ that satisfy both equations.

Solving by Substitution

One effective method is substitution. From Equation (2): $$ xy = 3 \quad \Rightarrow \quad x = \frac{3}{y} \quad \text{(3)} $$ Substitute Equation (3) into Equation (1): $$ \left(\frac{3}{y}\right)y^2 = 4 \\ 3y = 4 \\ y = \frac{4}{3} $$ Once $y$ is determined, substitute back into Equation (3) to find $x$: $$ x = \frac{3}{\frac{4}{3}} = \frac{9}{4} $$ Therefore, the solution is $x = \frac{9}{4}$ and $y = \frac{4}{3}$.

Verification of Solutions

Always verify the solutions by substituting them back into the original equations: \begin{align*} xy &= \frac{9}{4} \times \frac{4}{3} = 3 \quad \checkmark \\ xy^2 &= \frac{9}{4} \times \left(\frac{4}{3}\right)^2 = \frac{9}{4} \times \frac{16}{9} = 4 \quad \checkmark \end{align*} Both equations are satisfied, confirming the solutions are correct.

Alternative Method: Elimination

Alternatively, elimination can be employed. Multiply Equation (2) by $y$ to align terms with Equation (1): $$ xy \times y = 3y \\ xy^2 = 3y \quad \text{(4)} $$ Now equate Equations (1) and (4): $$ 4 = 3y \\ y = \frac{4}{3} $$ Substituting back into Equation (2): $$ x \times \frac{4}{3} = 3 \\ x = \frac{9}{4} $$ The solutions remain consistent: $x = \frac{9}{4}$ and $y = \frac{4}{3}$.

Graphical Interpretation

Graphing the equations provides a visual perspective. Equation (2), $xy = 3$, represents a hyperbola, while Equation (1), $xy^2 = 4$, depicts a different hyperbolic curve. The intersection point of these curves corresponds to the solution of the system. Plotting them confirms the analytical solution derived earlier.

Applications of Nonlinear Systems

Solving nonlinear simultaneous equations is crucial in various real-world applications, including economics for optimizing functions, physics for modeling trajectories, and engineering for system designs. Mastery of these techniques equips students with the analytical skills needed for higher-level studies and professional problem-solving.

Advanced Concepts

Theoretical Foundations

Nonlinear simultaneous equations often require a deeper understanding of algebraic structures and functions. The interaction between different degrees of variables introduces complexities not present in linear systems. Fundamental principles such as the existence and uniqueness of solutions, based on the nature of the equations, are essential for advanced problem-solving.

Mathematical Derivations and Proofs

Deriving solutions for nonlinear systems involves manipulating equations to isolate variables, applying algebraic identities, and sometimes using iterative methods for more complex systems. Proofs often demonstrate the existence of solutions under specific conditions, leveraging theorems from algebra and calculus.

Complex Problem-Solving Strategies

Advanced techniques include:

• Newton-Raphson Method: An iterative approach for approximating roots of equations.
• Resultant Method: Eliminating one variable to form a single equation.
• Graphical Methods: Utilizing computational tools for precise graph plotting.

Interdisciplinary Connections

Nonlinear systems are pervasive across disciplines:

• Physics: Modeling planetary motions and electromagnetic fields.
• Biology: Describing population dynamics and enzyme kinetics.
• Economics: Analyzing supply and demand equilibrium.

Challenging Problems and Solutions

Consider a more complex system: $$ \begin{align} x^2 + y^2 &= 25 \quad \text{(5)} \\ xy &= 12 \quad \text{(6)} \end{align} $$ Solving this system: From Equation (6), $x = \frac{12}{y}$. Substitute into Equation (5): $$ \left(\frac{12}{y}\right)^2 + y^2 = 25 \\ \frac{144}{y^2} + y^2 = 25 \\ 144 + y^4 = 25y^2 \\ y^4 - 25y^2 + 144 = 0 $$ Let $z = y^2$: $$ z^2 - 25z + 144 = 0 \\ z = \frac{25 \pm \sqrt{625 - 576}}{2} = \frac{25 \pm \sqrt{49}}{2} \\ z = \frac{25 \pm 7}{2} \\ z = 16 \quad \text{or} \quad z = 9 $$ Thus, $y^2 = 16 \Rightarrow y = \pm4$ and $y^2 = 9 \Rightarrow y = \pm3$. Corresponding $x$ values: $$ x = \frac{12}{4} = 3, \quad x = \frac{12}{-4} = -3 \\ x = \frac{12}{3} = 4, \quad x = \frac{12}{-3} = -4 $$ Solutions: $(3,4)$, $(-3,-4)$, $(4,3)$, $(-4,-3)$.

Numerical Methods for Nonlinear Systems

For systems where analytical solutions are cumbersome or impossible, numerical methods become invaluable. Techniques such as fixed-point iteration, the Newton-Raphson method, and computational algorithms facilitate finding approximate solutions with high precision. These methods are integral in fields requiring complex simulations and real-time problem-solving.

Comparison Table

Summary and Key Takeaways