Problem 48 - College Algebra

Question

Find the solutions to the nonlinear equations with two variables. $$ \begin{array}{r} \frac{4}{x^{2}}+\frac{1}{y^{2}}=24 \\ \frac{5}{x^{2}}-\frac{2}{y^{2}}+4=0 \end{array} $$

Step-by-Step Solution

Answer

The solutions are complex. For \( y \) values of \( \pm \frac{\sqrt{13}}{5} \), calculate corresponding \( x \) values to satisfy both equations.

Step 1: Manipulate the First Equation

Consider the first equation: \( \frac{4}{x^2}+\frac{1}{y^2}=24 \). Rewrite it as \( \frac{4}{x^2} = 24 - \frac{1}{y^2} \). Thus, \( \frac{4}{x^2} = \frac{24y^2-1}{y^2} \).

Step 2: Rewrite the Second Equation

Now focus on the second equation: \( \frac{5}{x^2} - \frac{2}{y^2} + 4 = 0 \). Rearrange it to get \( \frac{5}{x^2} = \frac{2}{y^2} - 4 \). This simplifies further to \( \frac{5}{x^2} = \frac{2-4y^2}{y^2} \).

Step 3: Equalize the Expressions for \( \frac{1}{x^2} \)

Since both equations give expressions for \( \frac{1}{x^2} \), equate them: \( \frac{4}{24y^2-1} = \frac{5}{2-4y^2} \). Cross-multiply to solve for \( y^2 \): \( 4(2-4y^2) = 5(24y^2-1) \).

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

Expand and simplify the expression: \( 8 - 16y^2 = 120y^2 - 5 \). Rearrange to get \( 125y^2 = 13 \). Solve for \( y^2 \) by dividing both sides by 125: \( y^2 = \frac{13}{125} \). Thus, \( y = \pm \frac{\sqrt{13}}{5} \).

Step 5: Substitute \( y \) into the First Equation to Find \( x^2 \)

Substitute \( y^2 = \frac{13}{125} \) into the first equation. This gives \( \frac{4}{x^2} + \frac{1}{\frac{13}{125}} = 24 \). Simplify to find \( \frac{4}{x^2} = \frac{24 \times 13}{125} - 1 \). Solve \( \frac{4}{x^2} = \frac{287}{125} \), so \( x^2 = \frac{500}{287} \). Thus, \( x = \pm \frac{\sqrt{500}}{\sqrt{287}} \).

Step 6: Verify the Solutions

To ensure the correctness of these solutions, substitute back into the original equations. The values for \( x \) and \( y \) should satisfy both equations. Given the complexity, ensure calculations are precise by rechecking each substitution and computation.

Key Concepts

Two variable systems

In mathematics, a two-variable system involves finding solutions for equations that include two different variables. This concept is pivotal in solving systems of equations, especially nonlinear ones, where the relationship between the variables isn't directly proportional or doesn’t follow a straight line. Understanding this concept helps you tackle more complex mathematical problems.

When dealing with two-variable systems, you're essentially working with pairs of equations that share two variables. These systems are represented as:

• Simultaneous equations
• Nonlinear terms or variables raised to a power.

In our given problem, we have a system represented by two nonlinear equations. These involve fractions and variables raised to the power of two, making direct calculation a bit trickier. The central goal is to find values of \( x \) and \( y \) that satisfy both equations simultaneously.

Two-variable systems can often be solved using substitution or elimination methods, but when nonlinearity comes into play, algebraic manipulation becomes essential to arrive at the necessary solutions.

Solving equations

Solving equations involves finding the values of variables that make the equation true. A crucial skill for any math student, it requires understanding the relationships and properties of mathematical expressions.

For nonlinear systems, solving equations is more complex than linear ones due to the intricate nature and multiple possible solutions. In the given exercise, solving begins with isolating variables and expressions. The method involves:

• Rewriting the equations to express one variable in terms of the other. This simplifies handling the nonlinearity.
• Equating two expressions to find values that satisfy both conditions.

In our problem, both equations are manipulated to solve for \( \frac{1}{x^2} \). This involves simplifying each equation separately and then setting their results equal to one another. Solving requires careful consideration of all steps and precise arithmetic to uncover the correct values of \( y^2 \) and ultimately, \( x^2 \) once substitution is made.

As you solve these equations, remember the need for accuracy at each stage to ensure valid solutions.

Algebraic manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to make them easier to solve. With nonlinear equations, this skill becomes particularly critical to breaking down complex expressions.

In our nonlinear system problem, algebraic manipulation allows us to transform the equations into more usable forms:

• Rewriting fractions by isolating terms, as with \( \frac{4}{x^2} \) and \( \frac{5}{x^2} \).
• Cross-multiplying to eliminate fractions, allowing a clearer pathway to solve for the desired variable.
• Simplifying the expanded equations to find a common form or solution.

Algebraic manipulation here includes rearranging each equation to isolate \( \frac{1}{x^2} \) and \( \frac{1}{y^2} \), which helps equate the two expressions effectively. Moreover, during the manipulation, keeping track of arithmetic is essential to avoid simple errors that can lead to incorrect solutions.

Being adept at algebraic manipulation ensures that complex nonlinear systems are approached efficiently and correctly, resulting in accurate solutions.