Given y*z = -v*x, x + b*y + c = 0, z + e*v + f = 0, y - v = a*(x - z), evaluate x, y, z, v.
You asked:
Given \( y z = -v \cdot x \), \( x + b y + c = 0 \), \( z + e v + f = 0 \), \( y - v = a \cdot \left(x - z\right) \), evaluate \( x \), \( y \), \( z \), \( v \).
MathBot Answer:
The system of equations has infinitely many solutions.
$$\begin{aligned}x&=\frac{- b^{2} a f - a b c e - b c + b f - b \sqrt{a^{2} b^{2} f^{2} + 2 a^{2} b c f e + a^{2} c^{2} e^{2} + 6 a b c f - 2 f^{2} a b - 2 c^{2} a e + 6 a c f e + c^{2} + 2 c f + f^{2}} - 2 c e}{2 \left(b^{2} a + a b e + b + e\right)}\\y&=\frac{- 2 a b c + a b f - a c e - c - f + \sqrt{a^{2} b^{2} f^{2} + 2 a^{2} b c f e + a^{2} c^{2} e^{2} + 6 a b c f - 2 f^{2} a b - 2 c^{2} a e + 6 a c f e + c^{2} + 2 c f + f^{2}}}{2 \left(b^{2} a + a b e + b + e\right)}\\z&=- \frac{e \sqrt{a^{2} b^{2} f^{2} + 2 a^{2} b c f e + a^{2} c^{2} e^{2} + 6 a b c f - 2 f^{2} a b - 2 c^{2} a e + 6 a c f e + c^{2} + 2 c f + f^{2}}}{2 \left(a b e + a e^{2} + b + e\right)} - \frac{a b f e + a c e^{2} + 2 b f - c e + f e}{2 \left(b + e\right) \left(a e + 1\right)}\\v&=\frac{- a b f + a c e - 2 a f e - c - f + \sqrt{a^{2} b^{2} f^{2} + 2 a^{2} b c f e + a^{2} c^{2} e^{2} + 6 a b c f - 2 f^{2} a b - 2 c^{2} a e + 6 a c f e + c^{2} + 2 c f + f^{2}}}{2 \left(a b e + a e^{2} + b + e\right)}\end{aligned}$$