Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. $5y^2+5x^2=30$

Verified step by step guidance

1. Rewrite the given equation for clarity: \$5y^2 + 5x^2 = 30$.
2. To check symmetry with respect to the y-axis, replace \(x\) with \(-x\) and see if the equation remains unchanged: \$5y^2 + 5(-x)^2 = 30$.
3. To check symmetry with respect to the x-axis, replace \(y\) with \(-y\) and see if the equation remains unchanged: \$5(-y)^2 + 5x^2 = 30$.
4. To check symmetry with respect to the origin, replace both \(x\) with \(-x\) and \(y\) with \(-y\) and see if the equation remains unchanged: \$5(-y)^2 + 5(-x)^2 = 30$.
5. Compare the resulting equations from each substitution to the original equation to determine which symmetries hold true.

Key Concepts

Symmetry with Respect to the x-axis

A graph is symmetric about the x-axis if replacing y with -y in the equation yields an equivalent equation. This means the graph looks the same above and below the x-axis, reflecting points across it.

Symmetry with Respect to the y-axis

A graph is symmetric about the y-axis if replacing x with -x in the equation results in the same equation. This indicates the graph mirrors itself on the left and right sides of the y-axis.

Symmetry with Respect to the Origin

A graph is symmetric about the origin if replacing both x with -x and y with -y produces the original equation. This means the graph is unchanged when rotated 180 degrees around the origin.