To determine the symmetries of the equation [tex]\(3x^4 + 34y^4 = 49\)[/tex], we need to check for symmetry with respect to the [tex]\(x\)[/tex]-axis, [tex]\(y\)[/tex]-axis, and the origin. Let's go through it step-by-step:
### Check for symmetry with respect to the [tex]\(x\)[/tex]-axis
To check if the equation is symmetric with respect to the [tex]\(x\)[/tex]-axis, we need to replace [tex]\(y\)[/tex] with [tex]\(-y\)[/tex] and see if the equation remains unchanged.
Original equation:
[tex]\[ 3x^4 + 34y^4 = 49 \][/tex]
Replace [tex]\(y\)[/tex] with [tex]\(-y\)[/tex]:
[tex]\[ 3x^4 + 34(-y)^4 = 49 \][/tex]
Simplify [tex]\((-y)^4\)[/tex] which is [tex]\((y^4)\)[/tex]:
[tex]\[ 3x^4 + 34y^4 = 49 \][/tex]
The equation remains the same:
[tex]\[ \text{Symmetric with respect to the \(x\)-axis} \][/tex]
### Check for symmetry with respect to the [tex]\(y\)[/tex]-axis
To check if the equation is symmetric with respect to the [tex]\(y\)[/tex]-axis, we need to replace [tex]\(x\)[/tex] with [tex]\(-x\)[/tex] and see if the equation remains unchanged.
Original equation:
[tex]\[ 3x^4 + 34y^4 = 49 \][/tex]
Replace [tex]\(x\)[/tex] with [tex]\(-x\)[/tex]:
[tex]\[ 3(-x)^4 + 34y^4 = 49 \][/tex]
Simplify [tex]\((-x)^4\)[/tex] which is [tex]\((x^4)\)[/tex]:
[tex]\[ 3x^4 + 34y^4 = 49 \][/tex]
The equation remains the same:
[tex]\[ \text{Symmetric with respect to the \(y\)-axis} \][/tex]
### Check for symmetry with respect to the origin
To check if the equation is symmetric with respect to the origin, we need to replace both [tex]\(x\)[/tex] with [tex]\(-x\)[/tex] and [tex]\(y\)[/tex] with [tex]\(-y\)[/tex] and see if the equation remains unchanged.
Original equation:
[tex]\[ 3x^4 + 34y^4 = 49 \][/tex]
Replace [tex]\(x\)[/tex] with [tex]\(-x\)[/tex] and [tex]\(y\)[/tex] with [tex]\(-y\)[/tex]:
[tex]\[ 3(-x)^4 + 34(-y)^4 = 49 \][/tex]
Simplify [tex]\((-x)^4\)[/tex] and [tex]\((-y)^4\)[/tex] which are [tex]\((x^4)\)[/tex] and [tex]\((y^4)\)[/tex] respectively:
[tex]\[ 3x^4 + 34y^4 = 49 \][/tex]
The equation remains the same:
[tex]\[ \text{Symmetric with respect to the origin} \][/tex]
### Conclusion
The equation [tex]\(3x^4 + 34y^4 = 49\)[/tex] is symmetric with respect to the [tex]\(x\)[/tex]-axis, [tex]\(y\)[/tex]-axis, and the origin. Therefore, the correct answer is:
[tex]\[(1, 1, 1)\][/tex]
