Chapter 7.5, Problem 56E

To determine

The system of inequalities which describes the region.

Answer to Problem 56E

The system of equation is,

  $\{\begin{array}{l}{x}^2+y^2\le 16\\ y\le x\\ x\le 0\\ y\ge 0\end{array}$

Explanation of Solution

Given information:

  

Formula used:

  $\begin{array}{l}{x}^2+y^2=r\\ m_{1}x+c=y\end{array}$

Calculation:

Consider the following region,

  

According to the graph, an equation of a quarter part of a circle with (0,0) as its center along with an equation of line will fit the graph; assume the equations to be,

  $\begin{array}{l}{x}^2+y^2=r\\ m_{1}x+c=y\end{array}$

From the graph, it is clear that (4, 0) satisfies the quadratic equation. Hence, put (4, 0) in the equation,

  $\begin{array}{l}{x}^2+y^2=r\\ 4^2+0^2=r\\ r=16\end{array}$

Hence, the equation of the graph is $x^2+y^2=16$

From the graph, it is clear that (0, 0) and $\left(\sqrt{8},\sqrt{8}\right)$ satisfy the equation of the line.

Hence, put (0, 0) and $\left(\sqrt{8},\sqrt{8}\right)$ in the equation,

  $\begin{array}{l}y=m_{1}x+c\\ 0=m_1\left(0\right)+c_1\\ \sqrt{8}=m_1\left(\sqrt{8}\right)+c_1\end{array}$

Solve the equations, and obtain the below,

  $\begin{array}{l}{m}_1=1\\ c_1=0\end{array}$

Hence, the equation of the line is y = x

The above shaded region is within the circle including it also and towards the origin hence symbol will be 5, for the line the region is away from the origin and the line itself is included so the symbol will be 2, and also the region lies only in the first quadrant where x and y are non- negative, hence the system of equation is,

  $\{\begin{array}{l}{x}^2+y^2\le 16\\ y\le x\\ x\le 0\\ y\ge 0\end{array}$

Conclusion:

The system of equation is,

  $\{\begin{array}{l}{x}^2+y^2\le 16\\ y\le x\\ x\le 0\\ y\ge 0\end{array}$