Solve u=varphi(y/x)+xpsi(y/x)

$$u = \varphi ( \frac { y } { x } ) + x \psi ( \frac { y } { x } )$$

Solve for x

$$\left\{\begin{matrix}x=\frac{y\phi }{u-y\psi }\text{, }&\phi \neq 0\text{ and }y\neq 0\text{ and }u\neq y\psi \\x\neq 0\text{, }&\left(u=0\text{ and }y=0\right)\text{ or }\left(\phi =0\text{ and }u=y\psi \right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$ux=\phi y+x\psi y$$

Subtract x\psi y from both sides.

$$ux-x\psi y=\phi y$$

Reorder the terms.

$$-xy\psi +ux=y\phi$$

Combine all terms containing x.

$$\left(-y\psi +u\right)x=y\phi$$

The equation is in standard form.

$$\left(u-y\psi \right)x=y\phi$$

Divide both sides by -y\psi +u.

$$\frac{\left(u-y\psi \right)x}{u-y\psi }=\frac{y\phi }{u-y\psi }$$

Dividing by -y\psi +u undoes the multiplication by -y\psi +u.

$$x=\frac{y\phi }{u-y\psi }$$

Variable x cannot be equal to 0.

$$x=\frac{y\phi }{u-y\psi }\text{, }x\neq 0$$

Solve for u

$$u=\frac{y\left(x\psi +\phi \right)}{x}$$

$$x\neq 0$$