Test for symmetry with respect to the x-axis and the y-axis. 
4y^(2)+5x^(2)=20

To test the equation \( 4y^2 + 5x^2 = 20 \) for symmetry with respect to the \(x\)-axis and the \(y\)-axis, we will follow these steps:

### Symmetry with respect to the \(x\)-axis
1. **Replace \(y\) with \(-y\)**:
\[ 4(-y)^2 + 5x^2 = 20 \]

2. Simplify the equation:
\[ 4y^2 + 5x^2 = 20 \]

Since the equation \( 4y^2 + 5x^2 = 20 \) is unchanged when \( y \) is replaced with \( -y \), the equation is symmetric with respect to the \(x\)-axis.

### Symmetry with respect to the \(y\)-axis
1. **Replace \(x\) with \(-x\)**:
\[ 4y^2 + 5(-x)^2 = 20 \]

2. Simplify the equation:
\[ 4y^2 + 5x^2 = 20 \]

Since the equation \( 4y^2 + 5x^2 = 20 \) is unchanged when \( x \) is replaced with \( -x \), the equation is symmetric with respect to the \(y\)-axis.

### Conclusion
The equation \( 4y^2 + 5x^2 = 20 \) is symmetric with respect to both the \(x\)-axis and the \(y\)-axis.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you test for symmetry with respect to the origin?
2. What does symmetry imply about the graph of an equation?
3. Can you give an example of an equation that is symmetric with respect to the origin but not the axes?
4. How does symmetry help in graphing equations?
5. What are the different types of symmetries that can be found in equations?

**Tip:** When testing for symmetry, always replace the variable and simplify the equation to see if it remains unchanged. This will help determine the type of symmetry present.

Test for symmetry with respect to the x-axis and the y-axis. 4y^2 + 5x^2 = 20

Learn how to test for symmetry with respect to the x-axis and y-axis using the equation 4y^2 + 5x^2 = 20. This problem covers symmetry principles, conic sections, and ellipse equations, suitable for students in Grades 9-12.

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