Solve z=1/y-5xy

$$z= \frac{ 1 }{ y } -5xy$$

Solve for x

$$x=\frac{1-yz}{5y^{2}}$$

$$y\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

$$zy=1-5xyy$$

Multiply y and y to get y^{2}.

$$zy=1-5xy^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$1-5xy^{2}=zy$$

Subtract 1 from both sides.

$$-5xy^{2}=zy-1$$

The equation is in standard form.

$$\left(-5y^{2}\right)x=yz-1$$

Divide both sides by -5y^{2}.

$$\frac{\left(-5y^{2}\right)x}{-5y^{2}}=\frac{yz-1}{-5y^{2}}$$

Dividing by -5y^{2} undoes the multiplication by -5y^{2}.

$$x=\frac{yz-1}{-5y^{2}}$$

Divide zy-1 by -5y^{2}.

$$x=-\frac{yz-1}{5y^{2}}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{\sqrt{20x+z^{2}}-z}{10x}\text{; }y=-\frac{\sqrt{20x+z^{2}}+z}{10x}\text{, }&x\neq 0\\y=\frac{1}{z}\text{, }&x=0\text{ and }z\neq 0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{\sqrt{20x+z^{2}}+z}{10x}\text{, }&\left(x\geq -\frac{z^{2}}{20}\text{ and }z>0\text{ and }z\geq 2\sqrt{-5x}\text{ and }x<0\right)\text{ or }\left(|z|\geq 2\sqrt{-5x}\text{ and }x\geq -\frac{z^{2}}{20}\text{ and }x<0\right)\text{ or }x>0\\y=\frac{\sqrt{20x+z^{2}}-z}{10x}\text{, }&\left(x\geq -\frac{z^{2}}{20}\text{ and }z<0\text{ and }z\leq -2\sqrt{-5x}\text{ and }x<0\right)\text{ or }\left(|z|\geq 2\sqrt{-5x}\text{ and }x\geq -\frac{z^{2}}{20}\text{ and }x<0\right)\text{ or }x>0\\y=\frac{1}{z}\text{, }&x=0\text{ and }z\neq 0\end{matrix}\right.$$