Whakaoti i te 3x/5+4/x-2x/y=2

$$\frac { 3 x } { 5 } + \frac { 4 } { x } - \frac { 2 x } { y } = 2$$

Whakaoti mō y (complex solution)

$$y=-\frac{10x^{2}}{-3x^{2}+10x-20}$$

$$x\neq 0\text{ and }x\neq \frac{5+\sqrt{35}i}{3}\text{ and }x\neq \frac{-\sqrt{35}i+5}{3}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5xy, arā, te tauraro pātahi he tino iti rawa te kitea o 5,x,y.

$$xy\times 3x+5y\times 4-5x\times 2x=10xy$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}y\times 3+5y\times 4-5x\times 2x=10xy$$

Whakareatia te 5 ki te 4, ka 20.

$$x^{2}y\times 3+20y-5x\times 2x=10xy$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}y\times 3+20y-5x^{2}\times 2=10xy$$

Whakareatia te 5 ki te 2, ka 10.

$$x^{2}y\times 3+20y-10x^{2}=10xy$$

Tangohia te 10xy mai i ngā taha e rua.

$$x^{2}y\times 3+20y-10x^{2}-10xy=0$$

Me tāpiri te 10x^{2} ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}y\times 3+20y-10xy=10x^{2}$$

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

$$\left(x^{2}\times 3+20-10x\right)y=10x^{2}$$

He hanga arowhānui tō te whārite.

$$\left(3x^{2}-10x+20\right)y=10x^{2}$$

Whakawehea ngā taha e rua ki te 3x^{2}-10x+20.

$$\frac{\left(3x^{2}-10x+20\right)y}{3x^{2}-10x+20}=\frac{10x^{2}}{3x^{2}-10x+20}$$

Mā te whakawehe ki te 3x^{2}-10x+20 ka wetekia te whakareanga ki te 3x^{2}-10x+20.

$$y=\frac{10x^{2}}{3x^{2}-10x+20}$$

Tē taea kia ōrite te tāupe y ki 0.

$$y=\frac{10x^{2}}{3x^{2}-10x+20}\text{, }y\neq 0$$

Whakaoti mō y

$$y=-\frac{10x^{2}}{-3x^{2}+10x-20}$$

$$x\neq 0$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5xy, arā, te tauraro pātahi he tino iti rawa te kitea o 5,x,y.

$$xy\times 3x+5y\times 4-5x\times 2x=10xy$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}y\times 3+5y\times 4-5x\times 2x=10xy$$

Whakareatia te 5 ki te 4, ka 20.

$$x^{2}y\times 3+20y-5x\times 2x=10xy$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}y\times 3+20y-5x^{2}\times 2=10xy$$

Whakareatia te 5 ki te 2, ka 10.

$$x^{2}y\times 3+20y-10x^{2}=10xy$$

Tangohia te 10xy mai i ngā taha e rua.

$$x^{2}y\times 3+20y-10x^{2}-10xy=0$$

Me tāpiri te 10x^{2} ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}y\times 3+20y-10xy=10x^{2}$$

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

$$\left(x^{2}\times 3+20-10x\right)y=10x^{2}$$

He hanga arowhānui tō te whārite.

$$\left(3x^{2}-10x+20\right)y=10x^{2}$$

Whakawehea ngā taha e rua ki te 3x^{2}-10x+20.

$$\frac{\left(3x^{2}-10x+20\right)y}{3x^{2}-10x+20}=\frac{10x^{2}}{3x^{2}-10x+20}$$

Mā te whakawehe ki te 3x^{2}-10x+20 ka wetekia te whakareanga ki te 3x^{2}-10x+20.

$$y=\frac{10x^{2}}{3x^{2}-10x+20}$$

Tē taea kia ōrite te tāupe y ki 0.

$$y=\frac{10x^{2}}{3x^{2}-10x+20}\text{, }y\neq 0$$

Whakaoti mō x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{5}\left(\sqrt{y\left(40-7y\right)}+\sqrt{5}y\right)}{3y-10}\text{; }x=\frac{\sqrt{5}\left(-\sqrt{y\left(40-7y\right)}+\sqrt{5}y\right)}{3y-10}\text{, }&y\neq \frac{10}{3}\text{ and }y\neq 0\\x=2\text{, }&y=\frac{10}{3}\end{matrix}\right.$$

Whakaoti mō x

$$\left\{\begin{matrix}x=\frac{\sqrt{5y}\left(\sqrt{40-7y}+\sqrt{5y}\right)}{3y-10}\text{; }x=\frac{\sqrt{5y}\left(-\sqrt{40-7y}+\sqrt{5y}\right)}{3y-10}\text{, }&y\neq \frac{10}{3}\text{ and }y\leq \frac{40}{7}\text{ and }y>0\\x=2\text{, }&y=\frac{10}{3}\end{matrix}\right.$$