find the value of x^2+y^3+z^3-yz if x^2+y^2+z^2=83 and x+z=15
You asked:
Given the following equations: \[\begin{aligned}{x}^{2} + {y}^{2} + {z}^{2} &= 83\\ x + z &= 15\end{aligned}\] Evaluate \({x}^{2} + {y}^{3} + {z}^{3} - y z\).
MathBot Answer:
The system of equations has infinitely many solutions.
$$\begin{aligned}x &= 15 - z\\y &= \sqrt{- 2 z^{2} + 30 z - 142}\end{aligned}$$\(z\) is a free variable.
By substituting these values into the expression, we find that: $$\begin{aligned}x^{2} + y^{3} - y z + z^{3} &= z^{3} - z \sqrt{- 2 z^{2} + 30 z - 142} + \left(z - 15\right)^{2} + 2 \sqrt{2} \left(- z^{2} + 15 z - 71\right)^{\frac{3}{2}}\end{aligned}$$$$\begin{aligned}x^{2} + y^{3} - y z + z^{3} &= z^{3} + z \sqrt{- 2 z^{2} + 30 z - 142} + \left(z - 15\right)^{2} - 2 \sqrt{2} \left(- z^{2} + 15 z - 71\right)^{\frac{3}{2}}\end{aligned}$$