Whakaoti i te y=2y+3/3y-2

$$y= \frac{ 2y+3 }{ 3y-2 }$$

Whakaoti mō y

$$y = \frac{\sqrt{13} + 2}{3} \approx 1.868517092$$

$$y=\frac{2-\sqrt{13}}{3}\approx -0.535183758$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te \frac{2y+3}{3y-2} mai i ngā taha e rua.

$$y-\frac{2y+3}{3y-2}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia y ki te \frac{3y-2}{3y-2}.

$$\frac{y\left(3y-2\right)}{3y-2}-\frac{2y+3}{3y-2}=0$$

Tā te mea he rite te tauraro o \frac{y\left(3y-2\right)}{3y-2} me \frac{2y+3}{3y-2}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{y\left(3y-2\right)-\left(2y+3\right)}{3y-2}=0$$

Mahia ngā whakarea i roto o y\left(3y-2\right)-\left(2y+3\right).

$$\frac{3y^{2}-2y-2y-3}{3y-2}=0$$

Whakakotahitia ngā kupu rite i 3y^{2}-2y-2y-3.

$$\frac{3y^{2}-4y-3}{3y-2}=0$$

Tē taea kia ōrite te tāupe y ki \frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3y-2.

$$3y^{2}-4y-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -4 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Pūrua -4.

$$y=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-3\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$y=\frac{-\left(-4\right)±\sqrt{16-12\left(-3\right)}}{2\times 3}$$

Whakareatia -12 ki te -3.

$$y=\frac{-\left(-4\right)±\sqrt{16+36}}{2\times 3}$$

Tāpiri 16 ki te 36.

$$y=\frac{-\left(-4\right)±\sqrt{52}}{2\times 3}$$

Tuhia te pūtakerua o te 52.

$$y=\frac{-\left(-4\right)±2\sqrt{13}}{2\times 3}$$

Ko te tauaro o -4 ko 4.

$$y=\frac{4±2\sqrt{13}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$y=\frac{4±2\sqrt{13}}{6}$$

Nā, me whakaoti te whārite y=\frac{4±2\sqrt{13}}{6} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{13}.

$$y=\frac{2\sqrt{13}+4}{6}$$

Whakawehe 4+2\sqrt{13} ki te 6.

$$y=\frac{\sqrt{13}+2}{3}$$

Nā, me whakaoti te whārite y=\frac{4±2\sqrt{13}}{6} ina he tango te ±. Tango 2\sqrt{13} mai i 4.

$$y=\frac{4-2\sqrt{13}}{6}$$

Whakawehe 4-2\sqrt{13} ki te 6.

$$y=\frac{2-\sqrt{13}}{3}$$

Kua oti te whārite te whakatau.

$$y=\frac{\sqrt{13}+2}{3} y=\frac{2-\sqrt{13}}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te \frac{2y+3}{3y-2} mai i ngā taha e rua.

$$y-\frac{2y+3}{3y-2}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia y ki te \frac{3y-2}{3y-2}.

$$\frac{y\left(3y-2\right)}{3y-2}-\frac{2y+3}{3y-2}=0$$

Tā te mea he rite te tauraro o \frac{y\left(3y-2\right)}{3y-2} me \frac{2y+3}{3y-2}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{y\left(3y-2\right)-\left(2y+3\right)}{3y-2}=0$$

Mahia ngā whakarea i roto o y\left(3y-2\right)-\left(2y+3\right).

$$\frac{3y^{2}-2y-2y-3}{3y-2}=0$$

Whakakotahitia ngā kupu rite i 3y^{2}-2y-2y-3.

$$\frac{3y^{2}-4y-3}{3y-2}=0$$

Tē taea kia ōrite te tāupe y ki \frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3y-2.

$$3y^{2}-4y-3=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y^{2}-4y=3$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3y^{2}-4y}{3}=\frac{3}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$y^{2}-\frac{4}{3}y=\frac{3}{3}$$

Whakawehe 3 ki te 3.

$$y^{2}-\frac{4}{3}y=1$$

Whakawehea te -\frac{4}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{3}. Nā, tāpiria te pūrua o te -\frac{2}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{4}{3}y+\left(-\frac{2}{3}\right)^{2}=1+\left(-\frac{2}{3}\right)^{2}$$

Pūruatia -\frac{2}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{4}{3}y+\frac{4}{9}=1+\frac{4}{9}$$

Tāpiri 1 ki te \frac{4}{9}.

$$y^{2}-\frac{4}{3}y+\frac{4}{9}=\frac{13}{9}$$

Tauwehea y^{2}-\frac{4}{3}y+\frac{4}{9}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{2}{3}\right)^{2}=\frac{13}{9}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(y-\frac{2}{3}\right)^{2}}=\sqrt{\frac{13}{9}}$$

Whakarūnātia.

$$y-\frac{2}{3}=\frac{\sqrt{13}}{3} y-\frac{2}{3}=-\frac{\sqrt{13}}{3}$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$y=\frac{\sqrt{13}+2}{3} y=\frac{2-\sqrt{13}}{3}$$