Solve frac{operatorname{gy}x}{x+1}+x+1/x=13/6

$$\frac { \operatorname { gy } x } { x + 1 } + \frac { x + 1 } { x } = \frac { 13 } { 6 }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=\frac{\left(7x-6\right)\left(x+1\right)}{6yx^{2}}\text{, }&x\neq 0\text{ and }y\neq 0\text{ and }x\neq -1\\g\in \mathrm{C}\text{, }&x=\frac{6}{7}\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6x\left(x+1\right), the least common multiple of x+1,x,6.

$$6xgyx+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}gy+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

Use the distributive property to multiply 6x+6 by x+1 and combine like terms.

$$6x^{2}gy+6x^{2}+12x+6=13x\left(x+1\right)$$

Use the distributive property to multiply 13x by x+1.

$$6x^{2}gy+6x^{2}+12x+6=13x^{2}+13x$$

Subtract 6x^{2} from both sides.

$$6x^{2}gy+12x+6=13x^{2}+13x-6x^{2}$$

Combine 13x^{2} and -6x^{2} to get 7x^{2}.

$$6x^{2}gy+12x+6=7x^{2}+13x$$

Subtract 12x from both sides.

$$6x^{2}gy+6=7x^{2}+13x-12x$$

Combine 13x and -12x to get x.

$$6x^{2}gy+6=7x^{2}+x$$

Subtract 6 from both sides.

$$6x^{2}gy=7x^{2}+x-6$$

The equation is in standard form.

$$6yx^{2}g=7x^{2}+x-6$$

Divide both sides by 6x^{2}y.

$$\frac{6yx^{2}g}{6yx^{2}}=\frac{\left(7x-6\right)\left(x+1\right)}{6yx^{2}}$$

Dividing by 6x^{2}y undoes the multiplication by 6x^{2}y.

$$g=\frac{\left(7x-6\right)\left(x+1\right)}{6yx^{2}}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{\left(7x-6\right)\left(x+1\right)}{6yx^{2}}\text{, }&x\neq 0\text{ and }y\neq 0\text{ and }x\neq -1\\g\in \mathrm{R}\text{, }&x=\frac{6}{7}\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6x\left(x+1\right), the least common multiple of x+1,x,6.

$$6xgyx+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}gy+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

Use the distributive property to multiply 6x+6 by x+1 and combine like terms.

$$6x^{2}gy+6x^{2}+12x+6=13x\left(x+1\right)$$

Use the distributive property to multiply 13x by x+1.

$$6x^{2}gy+6x^{2}+12x+6=13x^{2}+13x$$

Subtract 6x^{2} from both sides.

$$6x^{2}gy+12x+6=13x^{2}+13x-6x^{2}$$

Combine 13x^{2} and -6x^{2} to get 7x^{2}.

$$6x^{2}gy+12x+6=7x^{2}+13x$$

Subtract 12x from both sides.

$$6x^{2}gy+6=7x^{2}+13x-12x$$

Combine 13x and -12x to get x.

$$6x^{2}gy+6=7x^{2}+x$$

Subtract 6 from both sides.

$$6x^{2}gy=7x^{2}+x-6$$

The equation is in standard form.

$$6yx^{2}g=7x^{2}+x-6$$

Divide both sides by 6x^{2}y.

$$\frac{6yx^{2}g}{6yx^{2}}=\frac{\left(7x-6\right)\left(x+1\right)}{6yx^{2}}$$

Dividing by 6x^{2}y undoes the multiplication by 6x^{2}y.

$$g=\frac{\left(7x-6\right)\left(x+1\right)}{6yx^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{169-144gy}-13}{12gy+\sqrt{169-144gy}-13}\text{, }&y\neq 0\text{ and }g\neq 0\\x=\frac{\sqrt{169-144gy}+13}{12gy-\sqrt{169-144gy}-13}\text{, }&g\neq \frac{7}{6y}\text{ and }y\neq 0\text{ and }g\neq 0\\x=\frac{6}{7}\text{, }&y=0\text{ or }g=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{169-144gy}-13}{12gy+\sqrt{169-144gy}-13}\text{, }&\left(g\neq 0\text{ and }g\geq \frac{169}{144y}\text{ and }y<0\right)\text{ or }\left(g\neq 0\text{ and }g\leq \frac{169}{144y}\text{ and }y>0\right)\text{ or }\left(g=\frac{169}{144y}\text{ and }y\neq 0\right)\\x=\frac{\sqrt{169-144gy}+13}{12gy-\sqrt{169-144gy}-13}\text{, }&\left(g\neq \frac{7}{6y}\text{ and }g\geq \frac{169}{144y}\text{ and }g\neq 0\text{ and }y<0\right)\text{ or }\left(g\neq \frac{7}{6y}\text{ and }g\leq \frac{169}{144y}\text{ and }g\neq 0\text{ and }y>0\right)\text{ or }\left(g=\frac{169}{144y}\text{ and }y\neq 0\right)\\x=12\text{, }&g=\frac{169}{144y}\text{ and }y\neq 0\\x=\frac{6}{7}\text{, }&y=0\text{ or }g=0\end{matrix}\right.$$