Question

Solve each system. Use any method you wish. $$ \left\\{\begin{array}{l} \frac{1}{x^{4}}-\frac{1}{y^{4}}=1 \\ \frac{1}{x^{4}}+\frac{1}{y^{4}}=4 \end{array}\right. $$

Short answer

x = \sqrt[4]{\frac{2}{5}}, y = \sqrt[4]{\frac{2}{3}}

Step-by-step solution

Define variables for simplification

Let’s define two new variables: \[ a = \frac{1}{x^4} \text{ and } b = \frac{1}{y^4} \]This simplifies the system of equations to:\[ a - b = 1 \]\[ a + b = 4 \]

Solve for one variable

Add the two equations to eliminate variable b:\[ (a - b) + (a + b) = 1 + 4 \]This simplifies to:\[ 2a = 5 \]Solve for a:\[ a = \frac{5}{2} \]

Solve for the other variable

Use the value of a to find b. Substitute \( a = \frac{5}{2} \) into the second equation:\[ \frac{5}{2} + b = 4 \]Solve for b:\[ b = 4 - \frac{5}{2} = \frac{8}{2} - \frac{5}{2} = \frac{3}{2} \]

Convert back to the original variables

Recall the original definitions:\[ \frac{1}{x^4} = a = \frac{5}{2} \]\[ \frac{1}{y^4} = b = \frac{3}{2} \]So,\[ x^4 = \frac{2}{5} \]\[ y^4 = \frac{2}{3} \]

Solve for x and y

Taking the fourth root of both sides:\[ x = \sqrt[4]{\frac{2}{5}} \]\[ y = \sqrt[4]{\frac{2}{3}} \]

Key concepts

Algebra Simplification

When solving systems of equations, simplification is a powerful tool. Simplifying complex equations makes it easier to find a solution. Key methods to simplify equations include:

- Redefining new variables to replace complex expressions
- Combining like terms
- Using fundamental algebraic operations like addition, subtraction, multiplication, and division

In our exercise, the given system of equations is:\[ \frac{1}{x^{4}}-\frac{1}{y^{4}}=1 \] \[ \frac{1}{x^{4}}+\frac{1}{y^{4}}=4 \].

We simplify by defining two new variables: \[ a = \frac{1}{x^4} \text{ and } b = \frac{1}{y^4} \]. This transforms the original complex system into a simpler one: \[ a - b = 1 \]\[ a + b = 4 \]. This makes it much easier to handle.

Variable Substitution

Variable substitution is a crucial step when simplifying and solving equations. It involves replacing one variable with another already defined or substituting one part of the equation to simplify further steps. In our example, we redefined the variables to make the equations easier to work with.

We substituted \[ a = \frac{1}{x^4} \] and \[ b = \frac{1}{y^4} \]. By substituting these new variables into the equations, we simplified the complexity.

Doing so allowed us to manipulate the simpler forms of the equations, making our calculations more manageable and straightforward.

Solving for Variables

Once we have simplified the system and substituted the variables, the final step is to solve for the unknowns. For instance, we first combined the equations to eliminate one variable:

\[ (a - b) + (a + b) = 1 + 4 \] \[ 2a = 5 \].

We then solved for \[ a \]: \[ a = \frac{5}{2} \]. With \[ a \] known, we substituted back to find \[ b \]: \[ \frac{5}{2} + b = 4 \]\[ b = \frac{3}{2} \].

After finding \[ a \] and \[ b \], we reverted to the original variables by solving:

\[ x^4 = \frac{2}{5} \]

\[ y^4 = \frac{2}{3} \], \[ x = \frac{\text{root}}{4}{\frac{2}{5}} \]\[ y = \frac{\text{root}}{4}{\frac{2}{3}} \].

In conclusion, solving for variables involves reversing the substitution and solving the final values of the original unknowns.