Solve z/x+1+sz/x=xs+s/xg

$$\frac { z } { x + 1 } + \frac { s z } { x } = x s + \frac { s } { x g }$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{s\left(x+1\right)}{sx^{3}+sx^{2}-sxz-xz-sz}\text{, }&\left(x\neq -1\text{ and }s\neq 0\text{ and }x\neq 0\text{ and }z\neq \frac{s\left(x+1\right)x^{2}}{sx+x+s}\text{ and }z\neq \left(x+1\right)x^{2}\right)\text{ or }\left(x\neq -1\text{ and }s\neq 0\text{ and }x\neq 0\text{ and }z\neq \frac{s\left(x+1\right)x^{2}}{sx+x+s}\text{ and }s\neq -1\right)\text{ or }\left(x=-\frac{s}{s+1}\text{ and }s\neq 0\text{ and }s\neq -1\right)\\g\neq 0\text{, }&x\neq -1\text{ and }x\neq 0\text{ and }s=0\text{ and }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable g cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by gx\left(x+1\right), the least common multiple of x+1,x,xg.

$$gxz+\left(gx+g\right)sz=xsgx\left(x+1\right)+\left(x+1\right)s$$

Use the distributive property to multiply gx+g by s.

$$gxz+\left(gxs+gs\right)z=xsgx\left(x+1\right)+\left(x+1\right)s$$

Use the distributive property to multiply gxs+gs by z.

$$gxz+gxsz+gsz=xsgx\left(x+1\right)+\left(x+1\right)s$$

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

$$gxz+gxsz+gsz=x^{2}sg\left(x+1\right)+\left(x+1\right)s$$

Use the distributive property to multiply x^{2}sg by x+1.

$$gxz+gxsz+gsz=sgx^{3}+x^{2}sg+\left(x+1\right)s$$

Use the distributive property to multiply x+1 by s.

$$gxz+gxsz+gsz=sgx^{3}+x^{2}sg+xs+s$$

Subtract sgx^{3} from both sides.

$$gxz+gxsz+gsz-sgx^{3}=x^{2}sg+xs+s$$

Subtract x^{2}sg from both sides.

$$gxz+gxsz+gsz-sgx^{3}-x^{2}sg=xs+s$$

Reorder the terms.

$$-gsx^{3}-gsx^{2}+gsxz+gxz+gsz=sx+s$$

Combine all terms containing g.

$$\left(-sx^{3}-sx^{2}+sxz+xz+sz\right)g=sx+s$$

The equation is in standard form.

$$\left(sz+xz+sxz-sx^{2}-sx^{3}\right)g=sx+s$$

Divide both sides by zsx+zx+zs-sx^{3}-sx^{2}.

$$\frac{\left(sz+xz+sxz-sx^{2}-sx^{3}\right)g}{sz+xz+sxz-sx^{2}-sx^{3}}=\frac{sx+s}{sz+xz+sxz-sx^{2}-sx^{3}}$$

Dividing by zsx+zx+zs-sx^{3}-sx^{2} undoes the multiplication by zsx+zx+zs-sx^{3}-sx^{2}.

$$g=\frac{sx+s}{sz+xz+sxz-sx^{2}-sx^{3}}$$

Divide sx+s by zsx+zx+zs-sx^{3}-sx^{2}.

$$g=\frac{s\left(x+1\right)}{sz+xz+sxz-sx^{2}-sx^{3}}$$

Variable g cannot be equal to 0.

$$g=\frac{s\left(x+1\right)}{sz+xz+sxz-sx^{2}-sx^{3}}\text{, }g\neq 0$$

Solve for s

$$\left\{\begin{matrix}s=\frac{gxz}{\left(x+1\right)\left(gx^{2}-gz+1\right)}\text{, }&x\neq -1\text{ and }z\neq x^{2}+\frac{1}{g}\text{ and }x\neq 0\text{ and }g\neq 0\\s\in \mathrm{R}\text{, }&g=-\frac{1}{x^{2}}\text{ and }z=0\text{ and }x\neq -1\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by gx\left(x+1\right), the least common multiple of x+1,x,xg.

$$gxz+\left(gx+g\right)sz=xsgx\left(x+1\right)+\left(x+1\right)s$$

Use the distributive property to multiply gx+g by s.

$$gxz+\left(gxs+gs\right)z=xsgx\left(x+1\right)+\left(x+1\right)s$$

Use the distributive property to multiply gxs+gs by z.

$$gxz+gxsz+gsz=xsgx\left(x+1\right)+\left(x+1\right)s$$

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

$$gxz+gxsz+gsz=x^{2}sg\left(x+1\right)+\left(x+1\right)s$$

Use the distributive property to multiply x^{2}sg by x+1.

$$gxz+gxsz+gsz=sgx^{3}+x^{2}sg+\left(x+1\right)s$$

Use the distributive property to multiply x+1 by s.

$$gxz+gxsz+gsz=sgx^{3}+x^{2}sg+xs+s$$

Subtract sgx^{3} from both sides.

$$gxz+gxsz+gsz-sgx^{3}=x^{2}sg+xs+s$$

Subtract x^{2}sg from both sides.

$$gxz+gxsz+gsz-sgx^{3}-x^{2}sg=xs+s$$

Subtract xs from both sides.

$$gxz+gxsz+gsz-sgx^{3}-x^{2}sg-xs=s$$

Subtract s from both sides.

$$gxz+gxsz+gsz-sgx^{3}-x^{2}sg-xs-s=0$$

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

$$gxsz+gsz-sgx^{3}-x^{2}sg-xs-s=-gxz$$

Combine all terms containing s.

$$\left(gxz+gz-gx^{3}-x^{2}g-x-1\right)s=-gxz$$

The equation is in standard form.

$$\left(-gx^{3}-gx^{2}+gxz-x+gz-1\right)s=-gxz$$

Divide both sides by xzg-gx^{3}-x+zg-x^{2}g-1.

$$\frac{\left(-gx^{3}-gx^{2}+gxz-x+gz-1\right)s}{-gx^{3}-gx^{2}+gxz-x+gz-1}=-\frac{gxz}{-gx^{3}-gx^{2}+gxz-x+gz-1}$$

Dividing by xzg-gx^{3}-x+zg-x^{2}g-1 undoes the multiplication by xzg-gx^{3}-x+zg-x^{2}g-1.

$$s=-\frac{gxz}{-gx^{3}-gx^{2}+gxz-x+gz-1}$$

Divide -gxz by xzg-gx^{3}-x+zg-x^{2}g-1.

$$s=\frac{gxz}{\left(x+1\right)\left(gx^{2}-gz+1\right)}$$