Determine whether the graph is symmetric with respect to the \( x \)-axis, \( y \)-axis, and the origin. \( 6 x^{2}-9 y^{2}=7 \) Select all that apply. \( \square \) A. origin \( \square \) B. \( y \)-axis \( \square \) C. \( x \)-axis

To determine the symmetries of the graph given by the equation \( 6x^{2} - 9y^{2} = 7 \), we'll analyze each type of symmetry individually. ### 1. **Symmetry with Respect to the \( y \)-axis:** To test for \( y \)-axis symmetry, replace \( x \) with \( -x \) in the equation: \[ 6(-x)^{2} - 9y^{2} = 6x^{2} - 9y^{2} = 7 \] The equation remains unchanged, indicating that the graph is symmetric with respect to the \( y \)-axis. ### 2. **Symmetry with Respect to the \( x \)-axis:** To test for \( x \)-axis symmetry, replace \( y \) with \( -y \) in the equation: \[ 6x^{2} - 9(-y)^{2} = 6x^{2} - 9y^{2} = 7 \] Again, the equation remains unchanged, indicating symmetry with respect to the \( x \)-axis. ### 3. **Symmetry with Respect to the Origin:** To test for origin symmetry, replace both \( x \) with \( -x \) and \( y \) with \( -y \): \[ 6(-x)^{2} - 9(-y)^{2} = 6x^{2} - 9y^{2} = 7 \] The equation remains the same, confirming origin symmetry. ### **Conclusion:** All three symmetries apply to the graph of the equation \( 6x^{2} - 9y^{2} = 7 \). **Answer:** All of the above symmetries apply. - [x] A. origin - [x] B. \( y \)-axis - [x] C. \( x \)-axis