Question

consider the following system of equations.
\\(\

$$\begin{cases}- 10x^{2}-10y^{2}=-300\\6x^{2}+6y^{2}=150\\end{cases}$$

\\)
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:

First, rewrite the first equation \(-10x^{2}-10y^{2}=-300\) as \(x^{2}+y^{2}=30\) by dividing both sides by - 10. Rewrite the second - equation \(6x^{2}+6y^{2}=180\) as \(x^{2}+y^{2}=30\) by dividing both sides by 6. These are equations of circles in the standard form \((x - 0)^{2}+(y - 0)^{2}=(\sqrt{30})^{2}\). Since they are the same equation, they represent the same circle, and a system of equations representing the same graph has infinite solutions.

Answer:

The equations represent circles that result in the same graph.