Question: Based on the type of equations in the system, what is the greatest possible number of solutions?…

Based on the type of equations in the system, what is the greatest possible number of solutions?

$x^2 + y^2 = 9$

$9x + 2y = 16$

3

2

4

1

Solution

The problem involves a system of equations: \[ \begin{cases} x^2 + y^2 = 9 & \\ 9x + 2y = 16 & \end{cases} \] The first equation is a circle with a radius of 3, centered at the origin. The second equation is a line. We need to determine the greatest possible number of solutions, which is the number of intersection points between the circle and the line. A line can intersect a circle at most at 2 points. Therefore, the greatest possible number of solutions is 2.