11.40 How many real solutions does the system of equations have

$\left\{\begin{array}{l}x^{2}+y=5 \\ x+y^{2}=3 ?\end{array}\right.$
11.40 The solutions of the system are the coordinates of the points of intersection of the graphs of the functions given by these equations. Let's write the original system in the form

$\left\{\begin{array}{l}y=5-x^{2}, \\ x=3-y^{2} .\end{array}\right.$

The graph of the first equation is a parabola with the axis of symmetry $O y$, and the graph of the second equation is a parabola with the axis of symmetry $O x$. By plotting these graphs (Fig. 11.14), we obtain four points of their intersection, i.e., the system has four real solutions.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-361.jpg?height=563&width=518&top_left_y=330&top_left_x=166)

Fig. 11.14

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-361.jpg?height=560&width=464&top_left_y=324&top_left_x=691)

Fig. 11.15