Q. 60 - 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.

$\left\{\begin{array}{l}{x}^3+y^2=2\\ x^{2}y=4\end{array}\right.$

Step-by-Step Solution

Answer

The graph for the equation $\left\{\begin{array}{l}{x}^3+y^2=2\\ x^{2}y=4\end{array}\right.$ is,

From the graph, the solution of the given system is, $\left(-1.37,-2.14\right)$.

Step 1. Given information

$\left\{\begin{array}{l}{x}^3+y^2=2\\ x^{2}y=4\end{array}\right.$

First we need to solve each equations for $y$.

Let us consider the first equation.

$$\begin{gathered} x^3+y^2=2 \\ {y}^2=2-x^3 \\ y=\pm \sqrt{2-x^3} \end{gathered}$$

Next solve the second equation.

$$\begin{gathered} x^{2}y=4 \\ y=\frac{4}{x^2} \end{gathered}$$

Step 2 Now sketch a graph for the obtained equations.

In the graph, the two equations are intersecting at a point $\left(-1.37,2.14\right)$.

So, the system has only one solution.