Question

which graph shows the system \\( \

$$\begin{cases} x^{2}+y = 2 \\\\ x^{2}+y^{2}=9 \\end{cases}$$

\\)?

Explanation:

Step1: Analyze the first equation

The equation \(x^{2}+y = 2\) can be rewritten as \(y=-x^{2}+2\), which is a parabola opening downwards with vertex \((0,2)\).

Step2: Analyze the second equation

The equation \(x^{2}+y^{2}=9\) is a circle with center \((0,0)\) and radius \(r = 3\) (since \(r^{2}=9\)).

Step3: Visualize the intersection

We need to look for a graph that has a parabola \(y=-x^{2}+2\) and a circle \(x^{2}+y^{2}=9\) intersecting at some points.

Answer:

The graph that shows a downward - opening parabola \(y=-x^{2}+2\) and a circle \(x^{2}+y^{2}=9\) (center \((0,0)\), radius \(3\)) is the correct one.