To determine the maximum number of solutions to the given system of equations, let's analyze the types of equations involved and their geometric interpretations.
### Step-by-Step Analysis:
1. Identify the Equations:
[tex]\[
\begin{cases}
x^2 + y^2 = 9 \\
9x + 2y = 16
\end{cases}
\][/tex]
2. Type and Shape of Equations:
- The first equation [tex]\( x^2 + y^2 = 9 \)[/tex] represents a circle centered at the origin [tex]\((0,0)\)[/tex] with a radius of 3.
- The second equation [tex]\( 9x + 2y = 16 \)[/tex] represents a straight line.
3. Graphical Intersection:
- A circle and a straight line can intersect in several ways:
1. No Intersection: If the line is completely outside the circle.
2. One Intersection Point (Tangent): If the line just touches the circle at a single point.
3. Two Intersection Points: If the line passes through the circle, intersecting at two distinct points.
4. Greatest Possible Number of Solutions:
- The most intersections a line can have with a circle is two. This occurs when the line crosses through the circle.
Based on the shapes and possible intersections:
- The circle given by [tex]\( x^2 + y^2 = 9 \)[/tex] and the line described by [tex]\( 9x + 2y = 16 \)[/tex] can intersect at most in two points.
### Conclusion:
The greatest possible number of solutions to the given system of equations is:
[tex]\[
\boxed{2}
\][/tex]
