Firstly, are there always nontrivial real solutions to the sysytem of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$? [Answered]

Secondly, are there always nontrivial real solutions to the sysytem of equations, $A_{1}\frac{x^{5}}{\left|x\right|}+B_{1}\frac{y^{5}}{\left|y\right|}+C_{1}\frac{z^{5}}{\left|z\right|}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$? [Answered]

Thirdly, are there always nontrivial real solutions to the sysytem of equations, that $A_{1}\frac{x^{5}}{\left|x\right|}+B_{1}\frac{y^{5}}{\left|y\right|}+C_{1}\frac{z^{5}}{\left|z\right|}=0$ and $A_{2}\frac{x^{5}}{\left|x\right|}+B_{2}\frac{y^{5}}{\left|y\right|}+C_{2}\frac{z^{5}}{\left|z\right|}=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$?

Firstly, are there always nontrivial real solutions to the sysytem of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$? [Answered]

Secondly, are there always nontrivial real solutions to the sysytem of equations, $A_{1}\frac{x^{5}}{\left|x\right|}+B_{1}\frac{y^{5}}{\left|y\right|}+C_{1}\frac{z^{5}}{\left|z\right|}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$? [Answered]

Thirdly, are there always nontrivial real solutions to the sysytem of equations, that $A_{1}\frac{x^{5}}{\left|x\right|}+B_{1}\frac{y^{5}}{\left|y\right|}+C_{1}\frac{z^{5}}{\left|z\right|}=0$ and $A_{2}\frac{x^{5}}{\left|x\right|}+B_{2}\frac{y^{5}}{\left|y\right|}+C_{2}\frac{z^{5}}{\left|z\right|}=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$?

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Firstly, are there always nontrivial real solutions to the sysytem of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$, for real numbers $A_{1}$, $B_{1}$, $C_{1}$ be real numbers, not all zero, and C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$ be real numbersC_{2}$? [Answered]

Secondly, not all zero. Is it true that are there always nontrivial real solutions to the equation systemsysytem of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ A_{1}\frac{x^{5}}{\left|x\right|}+B_{1}\frac{y^{5}}{\left|y\right|}+C_{1}\frac{z^{5}}{\left|z\right|}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$, has a non-zero solution of A_{2}x+B_{2}y+C_{2}z=0$, for real numbers ?$A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, and $C_{2}$?

Let $A_{1}$, $B_{1}$, $C_{1}$ be real numbers, not all zero, and $A_{2}$, $B_{2}$, $C_{2}$ be real numbers, not all zero. Is it true that the equation system, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $ A_{2}x+B_{2}y+C_{2}z=0$, has a non-zero solution of real numbers?