Part 1: Graphing the Equations

The system of equations is:

1. $x^2 + y^2 = 9$
   This is the equation of a circle centered at the origin $(0, 0)$ with a radius of 3.

2. $x + y = 1$
   This is the equation of a straight line with a slope of $-1$ and a $y$-intercept of $1$.

Graphing the Equations:

The graph of the circle is a complete circle centered at the origin with points like $(3, 0)$, $(-3, 0)$, $(0, 3)$, and $(0, -3)$ on its perimeter.
The graph of the line $x + y = 1$ passes through points like $(1, 0)$ and $(0, 1)$.

Part 2: Finding the Number of Real Solutions

The real solutions are the intersection points of the circle and the line.

• The circle is symmetric, and the line will intersect it at exactly two distinct points.

Conclusion:

There are 2 real solutions to this system.

Would you like me to create and display the graph for better visualization?