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?
\\(\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 both simplified equations are of the form \(x^{2}+y^{2}=30\) (where \(a = 0\), \(b = 0\) and \(r=\sqrt{30}\)), so they represent circles. And since both equations simplify to the same equation, their graphs are the same circle, which means there are infinite solutions (all the points on the circle are solutions for both equations).

Now let's analyze the options:

• Option 1: The equations are circles (not parabolas) and their graphs are the same (so they intersect at all points on the circle), so this option is wrong.
• Option 2: The equations represent circles and their graphs are the same (so they intersect at infinite points), not non - intersecting, so this option is wrong.
• Option 3: The equations are circles (not parabolas), so this option is wrong.
• Option 4: The equations represent circles and after simplification, they are the same equation, so their graphs are the same circle, which means there are infinite solutions. This option is correct.

Answer:

D. The equations represent circles that result in the same graph.