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

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

Rešitev za g (complex solution)

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

Koraki za reševanje linearnih enačb

Pomnožite obe strani enačbe z 6x\left(x+1\right), najmanjšim skupnim mnogokratnikom števila x+1,x,6.

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

Pomnožite x in x, da dobite x^{2}.

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

Uporabite lastnost distributivnosti za množenje 6x+6 krat x+1 in kombiniranje pogojev podobnosti.

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

Uporabite distributivnost, da pomnožite 13x s/z x+1.

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

Odštejte 6x^{2} na obeh straneh.

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

Združite 13x^{2} in -6x^{2}, da dobite 7x^{2}.

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

Odštejte 12x na obeh straneh.

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

Združite 13x in -12x, da dobite x.

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

Odštejte 6 na obeh straneh.

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

Enačba je v standardni obliki.

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

Delite obe strani z vrednostjo 6x^{2}y.

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

Z deljenjem s/z 6x^{2}y razveljavite množenje s/z 6x^{2}y.

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

Rešitev za g

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

Koraki za reševanje linearnih enačb

Pomnožite obe strani enačbe z 6x\left(x+1\right), najmanjšim skupnim mnogokratnikom števila x+1,x,6.

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

Pomnožite x in x, da dobite x^{2}.

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

Uporabite lastnost distributivnosti za množenje 6x+6 krat x+1 in kombiniranje pogojev podobnosti.

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

Uporabite distributivnost, da pomnožite 13x s/z x+1.

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

Odštejte 6x^{2} na obeh straneh.

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

Združite 13x^{2} in -6x^{2}, da dobite 7x^{2}.

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

Odštejte 12x na obeh straneh.

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

Združite 13x in -12x, da dobite x.

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

Odštejte 6 na obeh straneh.

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

Enačba je v standardni obliki.

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

Delite obe strani z vrednostjo 6x^{2}y.

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

Z deljenjem s/z 6x^{2}y razveljavite množenje s/z 6x^{2}y.

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

Rešitev za x (complex solution)

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

Rešitev za x

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