Q. 64 - 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.

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

Step-by-Step Solution

Answer

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

From the graph, the solutions of the system are, $\left(0.13,-2.00\right),\left(1.89,0.64\right)$.

Step 1. Given information

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

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

Consider the first equation,

$$\begin{gathered} x^2+y^2=4 \\ {y}^2=4-x^2 \\ y=\pm \sqrt{4-x^2} \end{gathered}$$

Consider the second equation,

$y=\ln x$.

Step 2 Now sketch a graph for the obtained equations.

In the graph, the two equations are intersecting at two points. They are, $\left(1.89,0.64\right),\left(0.13,-2.00\right)$.

So, the system has two solutions.