Question

consider the following system of equations.
\

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

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

Explanation:

Step1: Simplify the first equation

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

Step2: Simplify the second equation

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

Step3: Analyze the type of equations and their graphs

The standard form of a circle is \((x - a)^{2}+(y - b)^{2}=r^{2}\), and our simplified equations are in the form \(x^{2}+y^{2}=r^{2}\) (where \(a = 0\), \(b = 0\) and \(r^{2}=30\)), so they represent circles. Since both equations simplify to the same equation \(x^{2}+y^{2}=30\), they result in the same graph. If two equations represent the same graph, the system has infinite solutions (all the points on the graph are solutions). Also, options about non - intersecting graphs are wrong because if graphs don't intersect, there are no solutions, and these are circles not parabolas.

Answer:

The equations represent circles that result in the same graph.