Question

consider the following system of equations.
\\( \left\\{ \

$$\begin{array} { r } { - 10 x ^ { 2 } - 10 y ^ { 2 } = - 300 } \\\\ { 5 x ^ { 2 } + 5 y ^ { 2 } = 150 } \\end{array}$$

\
ight. \\)
which statement describes why the system has infinite solutions?
the equations represent parabolas that result in graphs that do not intersect.
the equations represent circles that result in graphs that do not intersect.
the equations represent parabolas that result in the same graph.
the equations represent circles that result in the same graph.

Explanation:

Step1: Simplify the first equation

Divide the equation \(-10x^{2}-10y^{2}=-300\) by \(- 10\).
We get \(x^{2}+y^{2}=30\).

Step2: Simplify the second equation

Divide the equation \(5x^{2}+5y^{2}=150\) by \(5\).
We get \(x^{2}+y^{2}=30\).

The general equation of a circle is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \((a,b)\) is the center and \(r\) is the radius. In the case of \(x^{2}+y^{2}=30\), the center is \((0,0)\) and \(r=\sqrt{30}\). Since both equations simplify to the same circle equation \(x^{2}+y^{2}=30\), they represent the same circle.

Answer:

The equations represent circles that result in the same graph.