Question

which of the following graphs represents the solution(s) of the following system?
$x^{2}+y = 7$
$x^{2}+y^{2}=49$
done

Explanation:

Step1: Analyze the first equation \(x^{2}+y = 7\)

Rewrite it as \(y=-x^{2}+7\). This is a parabola that opens downwards (because the coefficient of \(x^{2}\) is \(- 1\)) with vertex at \((0,7)\).

Step2: Analyze the second equation \(x^{2}+y^{2}=49\)

This is a circle with center \((0,0)\) and radius \(r = 7\) (since the standard form of a circle is \((x - a)^{2}+(y - b)^{2}=r^{2}\), here \(a = 0\), \(b = 0\) and \(r^{2}=49\)).

Answer:

The graph with a downward - opening parabola \(y=-x^{2}+7\) and a circle \(x^{2}+y^{2}=49\) (the second graph from the left among the given options, assuming the order of analysis of the options: first is upward - opening parabola, second is downward - opening parabola and circle, third is some other non - matching combination, fourth is also non - matching) is the correct one.