Q. 62 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}{x}^4+y^4=6\\ xy=1\end{array}\right.$

Step-by-Step Solution

Answer

The graph for the system of equations $\left\{\begin{array}{l}{x}^4+y^4=6\\ xy=1\end{array}\right.$is,

From the graph, the solutions of the given system are,  $\left(0.64,1.56\right),\left(1.56,0.64\right),\left(-0.64,-1.56\right),\left(-1.56,-0.64\right)$.

Step 1. Given information

$\left\{\begin{array}{l}{x}^4+y^4=6\\ xy=1\end{array}\right.$

First solve the given equations for $y$.

Let us consider the first equation.

$$\begin{gathered} x^4+y^4=6 \\ {y}^4=6-x^4 \\ \ln \left(y^4\right)=\ln \left(6-x^4\right) \\ 4 \ln \left(y\right)=\ln \left(6-x^4\right) \\ \ln \left(y\right)=\frac{\ln \left(6-x^4\right)}{4} \\ y=e^{\frac{\ln \left(6-x^4\right)}{4}} \end{gathered}$$

Next consider the second equation.

$$\begin{gathered} xy=1 \\ y=\frac{1}{x} \end{gathered}$$

Step 2 Now sketch a graph for the obtained equations.

In the graph, the two equations are intersecting at four points. They are, $\left(0.64,1.56\right),\left(1.56,0.64\right),\left(-0.64,-1.56\right),\left(-1.56,-0.64\right)$. So, the system has four solutions.