Problem 36 - Algebra and Trigonometry

Question

Find all solutions of the system of equations. \(\left\\{\begin{array}{l}{\frac{4}{x^{2}}+\frac{6}{y^{4}}=\frac{7}{2}} \\\ {\frac{1}{x^{2}}-\frac{2}{y^{4}}=0}\end{array}\right.\)

Step-by-Step Solution

Answer

The solutions are \((\sqrt{2}, \sqrt{2}),\ (\sqrt{2}, -\sqrt{2}),\ (-\sqrt{2}, \sqrt{2}),\ (-\sqrt{2}, -\sqrt{2})\).

Step 1: Express \( y^4 \) in terms of \( x^2 \)

Start with the second equation: \( \frac{1}{x^2} - \frac{2}{y^4} = 0 \). Rearrange it to express \( y^4 \) in terms of \( x^2 \). Thus, \( \frac{1}{x^2} = \frac{2}{y^4} \). This simplifies to \( y^4 = 2x^2 \).

Step 2: Substitute \( y^4 = 2x^2 \) into the first equation

Take the expression for \( y^4 \) and substitute it into the first equation: \( \frac{4}{x^2} + \frac{6}{y^4} = \frac{7}{2} \). This becomes \( \frac{4}{x^2} + \frac{6}{2x^2} = \frac{7}{2} \). Simplify this to obtain \( \frac{4}{x^2} + \frac{3}{x^2} = \frac{7}{2} \).

Step 3: Simplify the equation

Combine the terms \( \frac{4}{x^2} + \frac{3}{x^2} \) to get \( \frac{7}{x^2} = \frac{7}{2} \).

Step 4: Solve for \( x^2 \)

Cross-multiply to solve for \( x^2 \): \( 14 = 7x^2 \). Divide both sides by 7 to find \( x^2 = 2 \).

Step 5: Solve for \( y^4 \) using \( x^2 = 2 \)

Substitute \( x^2 = 2 \) into the expression for \( y^4 \) which is \( y^4 = 2x^2 \). Thus, \( y^4 = 2*2 = 4 \).

Step 6: Determine \( x \) and \( y \) values

Since \( x^2 = 2 \), \( x \) can be either \( \sqrt{2} \) or \( -\sqrt{2} \). Similarly, since \( y^4 = 4 \), \( y^2 = \pm 2 \), and therefore \( y = \sqrt{2} \) or \( -\sqrt{2} \).

Step 7: List all solutions

The solutions are (x, y) pairs: \((\sqrt{2}, \sqrt{2}),\ (\sqrt{2}, -\sqrt{2}),\ (-\sqrt{2}, \sqrt{2}),\ (-\sqrt{2}, -\sqrt{2})\).

Key Concepts

Rational Expressions

Rational expressions are expressions that involve fractions with polynomials in the numerator and/or the denominator. They are quite similar to common fractions, but instead of integers, they involve variables.

Consider the expression \( \frac{4}{x^2} \). The polynomial \( x^2 \) is in the denominator, making the expression rational. Solving problems involving rational expressions often requires dealing with their domain, which is where the expression is defined. In this exercise, for \( \frac{4}{x^2} \) and other similar terms, we must ensure that \( x eq 0 \) to avoid division by zero.

When working with rational expressions in systems of equations, it’s critical to manage all common algebraic rules. This involves finding common denominators when adding fractions, simplifying expressions, and sometimes factoring numerators or denominators. In our original exercise, we combined rational expressions by finding a common denominator which simplified the solving process.

Equations Involving Radicals

Equations involving radicals include variables within a square root, cube root, or higher roots. Solving these equations generally involves eliminating the radical by raising both sides of the equation to the necessary power.

In our solution, this process appears indirectly when determining the values of \( x \) and \( y \) from their squared or another powered form. For instance, rather than handling radicals directly, we considered raising \( x \) and \( y \) to certain powers in steps to derive an equation solvable via more straightforward algebraic methods.

To effectively solve these equations, especially within systems, we should carefully follow algebraic principles to maintain equality. Incorrect manipulation, such as improper squaring or ignoring the domain constraints, could result in extraneous or incorrect solutions.

Cross-Multiplication

Cross-multiplication is a handy algebraic method used to solve equations involving two ratios set equal to each other, such as \( \frac{a}{b} = \frac{c}{d} \). The technique simplifies the expression to \( ad = bc \), allowing for easier manipulation and solution.

In the solution, cross-multiplication was employed to simplify \( \frac{7}{x^2} \) equaling \( \frac{7}{2} \). By cross-multiplying, the problem reduced to the linear equation \( 14 = 7x^2 \), from where we solved for \( x^2 \).

This concept is fundamental because it helps avoid the complexities often associated with fractions and inequalities, allowing a straightforward path to the solution. However, correct usage requires careful attention to the equality's terms to avoid mistakes, such as incorrectly simplifying terms or skipping necessary operations.