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

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

Resolver g (complex solution)

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

Pasos para resolver unha ecuación lineal

Multiplica ambos lados da ecuación por 6x\left(x+1\right), o mínimo común denominador de x+1,x,6.

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

Multiplica x e x para obter x^{2}.

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

Usa a propiedade distributiva para multiplicar 6x+6 por x+1 e combina os termos semellantes.

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

Usa a propiedade distributiva para multiplicar 13x por x+1.

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

Resta 6x^{2} en ambos lados.

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

Combina 13x^{2} e -6x^{2} para obter 7x^{2}.

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

Resta 12x en ambos lados.

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

Combina 13x e -12x para obter x.

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

Resta 6 en ambos lados.

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

A ecuación está en forma estándar.

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

Divide ambos lados entre 6x^{2}y.

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

A división entre 6x^{2}y desfai a multiplicación por 6x^{2}y.

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

Resolver g

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

Pasos para resolver unha ecuación lineal

Multiplica ambos lados da ecuación por 6x\left(x+1\right), o mínimo común denominador de x+1,x,6.

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

Multiplica x e x para obter x^{2}.

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

Usa a propiedade distributiva para multiplicar 6x+6 por x+1 e combina os termos semellantes.

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

Usa a propiedade distributiva para multiplicar 13x por x+1.

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

Resta 6x^{2} en ambos lados.

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

Combina 13x^{2} e -6x^{2} para obter 7x^{2}.

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

Resta 12x en ambos lados.

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

Combina 13x e -12x para obter x.

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

Resta 6 en ambos lados.

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

A ecuación está en forma estándar.

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

Divide ambos lados entre 6x^{2}y.

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

A división entre 6x^{2}y desfai a multiplicación por 6x^{2}y.

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

Resolver x (complex solution)

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

Resolver x

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