Problem 52 - College Algebra Essentials

Question

solve each system by the method of your choice. $$ \left\\{\begin{array}{l} \frac{2}{x^{2}}+\frac{1}{y^{2}}=11 \\ \frac{4}{x^{2}}-\frac{2}{y^{2}}=-14 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution of this system of equations requires a methodical approach that includes simplification, solving, and checking solutions.

Step 1: Rewrite the System of Equations

To make the system of equations easier to work with, it can be simplified by multiplying each term by \(x^{2}y^{2}\) to clear the fractions and make them quadratic equations. The system now becomes:\[\left\{\begin{array}{l}2y^{2}+x^{2}=11x^{2}y^{2} \4y^{2}-2x^{2}=-14x^{2}y^{2}\end{array}\right.\]

Step 2: Solving the Equations

To facilitate solving the system, the equations should be subtracted from each other. This gives:\[\begin{array}{l}2y^{2}+x^{2}-(4y^{2}-2x^{2})=11x^{2}y^{2}-(-14x^{2}y^{2})\-2y^{2}+3x^{2}=25x^{2}y^{2}\end{array}\]Dividing both sides by 25 to isolate \(x^{2}y^{2}\):\(x^{2}y^{2}\)=\(\frac{-2y^{2}+3x^{2}}{25}\)Substitute \(x^{2}y^{2}\) back to any of the two original equations, solely for x:2y^{2}+x^{2}=11(\(\frac{-2y^{2}+3x^{2}}{25}\))This equation can be solved for x using the quadratic formula.

Step 3: Check and Verify the Solutions

Substitute the obtained values of \(x\) and \(x^{2}y^{2}\) back into the original system of equations to make sure that both equations hold true. If they do, then the obtained solutions are correct.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. They usually appear in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In this exercise, we transform our system into quadratic equations to eliminate the fractions.Understanding the basics of quadratic equations is crucial because it allows us to find solutions by seeking the values of the variables that make the equation true. Quadratic equations can have zero, one, or two real solutions, depending on the discriminant \(b^2 - 4ac\).In our problem, we reformulate the given system to arrive at expressions that we can solve using methods applicable to quadratic equations, such as factoring or applying the quadratic formula. This transformation is a key step in solving the system.

Solving Systems

Solving systems of equations involves finding the set of variable values that satisfy all equations in the system. These systems can be linear or nonlinear, with our example being a nonlinear system because of the fractions and exponents present. Here, we start by manipulating the equations to form a system of quadratic equations, which simplifies the complexity usually associated with solving nonlinear systems. The goal is to reduce the system into a simpler form where the solution becomes more apparent. By rearranging and simplifying, we often subject one equation to operations such as addition, subtraction, or substitution using the other equation, helping us isolate variables to find solutions that satisfy both equations in the system.

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves solving one of the equations for one variable, then substituting this expression into the other equation.In the given exercise, after simplifying and transforming the equations, we isolate a critical expression, namely \(x^2y^2\), and substitute it back into one of the original equations. This substitution allows us to focus on solving for one variable at a time, simplifying a potentially complex system into more manageable single-variable problems.This method is particularly useful when one equation can easily be manipulated to express one variable in terms of another, which then streamlines the process of finding a solution to the whole system.

Verification of Solutions

Verification of solutions is a significant step in any mathematical problem involving systems of equations. Once potential solutions are found, it’s essential to substitute them back into the original equations to ensure they truly satisfy both equations.In the final step of our exercise, after calculating the possible values for the variables \(x\) and \(x^2y^2\), substituting these back into the original system is crucial. It checks the accuracy of the solutions by confirming each equation holds true under these values.Verification helps prevent errors from earlier steps and ensures a high degree of confidence in the correctness of the solutions, which is especially important with complex systems and transformations, such as quadratic equations.