Whakaoti i te r^2=x+1/x-1

$$r ^ { 2 } = \frac { x + 1 } { x - 1 }$$

Whakaoti mō x (complex solution)

$$x=\frac{r^{2}+1}{r^{2}-1}$$

$$r\neq -1\text{ and }r\neq 1$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-1.

$$\left(x-1\right)r^{2}=x+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te r^{2}.

$$xr^{2}-r^{2}=x+1$$

Tangohia te x mai i ngā taha e rua.

$$xr^{2}-r^{2}-x=1$$

Me tāpiri te r^{2} ki ngā taha e rua.

$$xr^{2}-x=1+r^{2}$$

Pahekotia ngā kīanga tau katoa e whai ana i te x.

$$\left(r^{2}-1\right)x=1+r^{2}$$

He hanga arowhānui tō te whārite.

$$\left(r^{2}-1\right)x=r^{2}+1$$

Whakawehea ngā taha e rua ki te r^{2}-1.

$$\frac{\left(r^{2}-1\right)x}{r^{2}-1}=\frac{r^{2}+1}{r^{2}-1}$$

Mā te whakawehe ki te r^{2}-1 ka wetekia te whakareanga ki te r^{2}-1.

$$x=\frac{r^{2}+1}{r^{2}-1}$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=\frac{r^{2}+1}{r^{2}-1}\text{, }x\neq 1$$

Whakaoti mō x

$$x=\frac{r^{2}+1}{r^{2}-1}$$

$$|r|\neq 1$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-1.

$$\left(x-1\right)r^{2}=x+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te r^{2}.

$$xr^{2}-r^{2}=x+1$$

Tangohia te x mai i ngā taha e rua.

$$xr^{2}-r^{2}-x=1$$

Me tāpiri te r^{2} ki ngā taha e rua.

$$xr^{2}-x=1+r^{2}$$

Pahekotia ngā kīanga tau katoa e whai ana i te x.

$$\left(r^{2}-1\right)x=1+r^{2}$$

He hanga arowhānui tō te whārite.

$$\left(r^{2}-1\right)x=r^{2}+1$$

Whakawehea ngā taha e rua ki te r^{2}-1.

$$\frac{\left(r^{2}-1\right)x}{r^{2}-1}=\frac{r^{2}+1}{r^{2}-1}$$

Mā te whakawehe ki te r^{2}-1 ka wetekia te whakareanga ki te r^{2}-1.

$$x=\frac{r^{2}+1}{r^{2}-1}$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=\frac{r^{2}+1}{r^{2}-1}\text{, }x\neq 1$$

Whakaoti mō r (complex solution)

$$r=-\left(x-1\right)^{-\frac{1}{2}}\sqrt{x+1}$$

$$r=\left(x-1\right)^{-\frac{1}{2}}\sqrt{x+1}\text{, }x\neq 1$$

Whakaoti mō r

$$r=\sqrt{\frac{x+1}{x-1}}$$

$$r=-\sqrt{\frac{x+1}{x-1}}\text{, }x>1\text{ or }x\leq -1$$