Q. 61 - 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.

$\left\{\begin{array}{l}{x}^4+y^4=12\\ x{y}^2=2\end{array}\right.$

Step-by-Step Solution

Answer

The graph of the system of equations $\left\{\begin{array}{l}{x}^4+y^4=12\\ x{y}^2=2\end{array}\right.$ is,

From the graph, the solutions of the given system are, $\left(1.81,1.05\right),\left(1.81,-1.05\right),\left(0.58,1.56\right),\left(0.58,-1.56\right)$.

Step 1. Given information

$\left\{\begin{array}{l}{x}^4+y^4=12\\ x{y}^2=2\end{array}\right.$

First we need to solve the given equations for $y$.

$$\begin{gathered} x^4+y^4=12 \\ {y}^4=12-x^4 \\ \ln \left(y^4\right)=\ln \left(12-x^4\right) \\ 4 \ln \left(y\right)=\ln \left(12-x^4\right) \\ \ln \left(y\right)=\frac{\ln \left(12-x^4\right)}{4} \\ y=e^{\frac{\ln \left(12-x^4\right)}{4}} \end{gathered}$$

Next solve the second equation.

$$\begin{gathered} x{y}^2=2 \\ {y}^2=\frac{2}{x} \\ y=\pm \sqrt{\frac{2}{x}} \end{gathered}$$

Step 2 Now sketch a graph for the obtained equations.

In the graph, the two equations are intersecting at four points. They are, $\left(1.81,1.05\right),\left(1.81,-1.05\right),\left(0.58,1.56\right),\left(0.58,-1.56\right)$.

So, the system has four solutions.