Chapter 9.7, Problem 1OE

To determine

To calculate: To find the number of real solutions of system of equations

Answer to Problem 1OE

Number of real solutions is $2$

Explanation of Solution

Given information: System of equations $x^2+y^2=4;y=x$

Calculation:

Given the system of equations as follows:

  $\begin{array}{l}{x}^2+y^2=4----(1)\\ y=x-------(2)\end{array}$

From equation (2)

  $y=x$

Substituting the value in (1):

  $\begin{array}{l}{x}^2+y^2=4\\ x^2+x^2=4\\ 2x^2=4\\ x^2=2\\ x=\pm \sqrt{2}\end{array}$

When $x=\sqrt{2}$

  $y=\sqrt{2}$

When $x=-\sqrt{2}$

  $y=-\sqrt{2}$

Thus, solutions are $\left(-\sqrt{2},-\sqrt{2}\right)$ and $\left(\sqrt{2},\sqrt{2}\right)$

Conclusion:

Hence, number of real solutions is $2$