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?

• 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:

Simplify the equations in the system

Divide the first equation by \(-10\) and the second equation by \(5\):

$$
LATEXBLOCK0
$$

Identify the geometric representation

Both simplified equations are of the form \(x^2 + y^2 = r^2\), which represents a circle centered at the origin \((0,0)\) with radius \(r = \sqrt{30}\).

Determine the relationship between the graphs

Since both equations simplify to the exact same equation \(x^2 + y^2 = 30\), they represent the same circle. Therefore, the graphs coincide completely, resulting in infinitely many intersection points (infinite solutions).

Answer:

• (A) The equations represent parabolas that result in graphs that do not intersect
• (B) The equations represent circles that result in graphs that do not intersect
• (C) The equations represent parabolas that result in the same graph
• (D) The equations represent circles that result in the same graph (Correct answer)