Question

consider the following system of equations.
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$$\begin{cases}- 10x^{2}-10y^{2}=-300 \\\\5x^{2}+5y^{2}=150\\end{cases}$$

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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 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 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 nature of the equations

The general form of a circle equation is $(x - a)^{2}+(y - b)^{2}=r^{2}$, and $x^{2}+y^{2}=30$ represents a circle centered at the origin $(0,0)$ with radius $r = \sqrt{30}$. Since the two - simplified equations are the same, they represent the same circle. So the system has infinite solutions.

Answer:

The equations represent circles that result in the same graph.