Solve 3xyp+zq+yz=0

$$3 x y p + z q + y z = 0$$

Solve for p

$$\left\{\begin{matrix}p=-\frac{z\left(y+q\right)}{3xy}\text{, }&y\neq 0\text{ and }x\neq 0\\p\in \mathrm{R}\text{, }&\left(z=0\text{ and }x=0\right)\text{ or }\left(z=0\text{ and }y=0\right)\text{ or }\left(y=-q\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }q=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract zq from both sides. Anything subtracted from zero gives its negation.

$$3xyp+yz=-zq$$

Subtract yz from both sides.

$$3xyp=-zq-yz$$

Reorder the terms.

$$3pxy=-yz-qz$$

The equation is in standard form.

$$3xyp=-yz-qz$$

Divide both sides by 3xy.

$$\frac{3xyp}{3xy}=-\frac{z\left(y+q\right)}{3xy}$$

Dividing by 3xy undoes the multiplication by 3xy.

$$p=-\frac{z\left(y+q\right)}{3xy}$$

Solve for q

$$\left\{\begin{matrix}q=-\frac{y\left(3px+z\right)}{z}\text{, }&z\neq 0\\q\in \mathrm{R}\text{, }&z=0\text{ and }\left(x=0\text{ or }p=0\text{ or }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 3xyp from both sides. Anything subtracted from zero gives its negation.

$$zq+yz=-3xyp$$

Subtract yz from both sides.

$$zq=-3xyp-yz$$

The equation is in standard form.

$$zq=-3pxy-yz$$

Divide both sides by z.

$$\frac{zq}{z}=-\frac{y\left(3px+z\right)}{z}$$

Dividing by z undoes the multiplication by z.

$$q=-\frac{y\left(3px+z\right)}{z}$$