Whakaoti i te 81/41-y+15=71/y

$$\frac { 81 } { 41 - y } + 15 = \frac { 71 } { y }$$

Whakaoti mō y

$$y = \frac{\sqrt{413629} + 767}{30} \approx 47.004665122$$

$$y = \frac{767 - \sqrt{413629}}{30} \approx 4.128668211$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe y ki tētahi o ngā uara 0,41 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te y\left(y-41\right), arā, te tauraro pātahi he tino iti rawa te kitea o 41-y,y.

$$-y\times 81+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Whakareatia te -1 ki te 81, ka -81.

$$-81y+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Whakamahia te āhuatanga tohatoha hei whakarea te y ki te y-41.

$$-81y+\left(y^{2}-41y\right)\times 15=\left(y-41\right)\times 71$$

Whakamahia te āhuatanga tohatoha hei whakarea te y^{2}-41y ki te 15.

$$-81y+15y^{2}-615y=\left(y-41\right)\times 71$$

Pahekotia te -81y me -615y, ka -696y.

$$-696y+15y^{2}=\left(y-41\right)\times 71$$

Whakamahia te āhuatanga tohatoha hei whakarea te y-41 ki te 71.

$$-696y+15y^{2}=71y-2911$$

Tangohia te 71y mai i ngā taha e rua.

$$-696y+15y^{2}-71y=-2911$$

Pahekotia te -696y me -71y, ka -767y.

$$-767y+15y^{2}=-2911$$

Me tāpiri te 2911 ki ngā taha e rua.

$$-767y+15y^{2}+2911=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$15y^{2}-767y+2911=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, -767 mō b, me 2911 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-767\right)±\sqrt{\left(-767\right)^{2}-4\times 15\times 2911}}{2\times 15}$$

Pūrua -767.

$$y=\frac{-\left(-767\right)±\sqrt{588289-4\times 15\times 2911}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$y=\frac{-\left(-767\right)±\sqrt{588289-60\times 2911}}{2\times 15}$$

Whakareatia -60 ki te 2911.

$$y=\frac{-\left(-767\right)±\sqrt{588289-174660}}{2\times 15}$$

Tāpiri 588289 ki te -174660.

$$y=\frac{-\left(-767\right)±\sqrt{413629}}{2\times 15}$$

Ko te tauaro o -767 ko 767.

$$y=\frac{767±\sqrt{413629}}{2\times 15}$$

Whakareatia 2 ki te 15.

$$y=\frac{767±\sqrt{413629}}{30}$$

Nā, me whakaoti te whārite y=\frac{767±\sqrt{413629}}{30} ina he tāpiri te ±. Tāpiri 767 ki te \sqrt{413629}.

$$y=\frac{\sqrt{413629}+767}{30}$$

Nā, me whakaoti te whārite y=\frac{767±\sqrt{413629}}{30} ina he tango te ±. Tango \sqrt{413629} mai i 767.

$$y=\frac{767-\sqrt{413629}}{30}$$

Kua oti te whārite te whakatau.

$$y=\frac{\sqrt{413629}+767}{30} y=\frac{767-\sqrt{413629}}{30}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe y ki tētahi o ngā uara 0,41 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te y\left(y-41\right), arā, te tauraro pātahi he tino iti rawa te kitea o 41-y,y.

$$-y\times 81+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Whakareatia te -1 ki te 81, ka -81.

$$-81y+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Whakamahia te āhuatanga tohatoha hei whakarea te y ki te y-41.

$$-81y+\left(y^{2}-41y\right)\times 15=\left(y-41\right)\times 71$$

Whakamahia te āhuatanga tohatoha hei whakarea te y^{2}-41y ki te 15.

$$-81y+15y^{2}-615y=\left(y-41\right)\times 71$$

Pahekotia te -81y me -615y, ka -696y.

$$-696y+15y^{2}=\left(y-41\right)\times 71$$

Whakamahia te āhuatanga tohatoha hei whakarea te y-41 ki te 71.

$$-696y+15y^{2}=71y-2911$$

Tangohia te 71y mai i ngā taha e rua.

$$-696y+15y^{2}-71y=-2911$$

Pahekotia te -696y me -71y, ka -767y.

$$-767y+15y^{2}=-2911$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$15y^{2}-767y=-2911$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15y^{2}-767y}{15}=-\frac{2911}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$y^{2}-\frac{767}{15}y=-\frac{2911}{15}$$

Whakawehea te -\frac{767}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{767}{30}. Nā, tāpiria te pūrua o te -\frac{767}{30} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{767}{15}y+\left(-\frac{767}{30}\right)^{2}=-\frac{2911}{15}+\left(-\frac{767}{30}\right)^{2}$$

Pūruatia -\frac{767}{30} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{767}{15}y+\frac{588289}{900}=-\frac{2911}{15}+\frac{588289}{900}$$

Tāpiri -\frac{2911}{15} ki te \frac{588289}{900} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y^{2}-\frac{767}{15}y+\frac{588289}{900}=\frac{413629}{900}$$

Tauwehea y^{2}-\frac{767}{15}y+\frac{588289}{900}. 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}.

$$\left(y-\frac{767}{30}\right)^{2}=\frac{413629}{900}$$

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

$$\sqrt{\left(y-\frac{767}{30}\right)^{2}}=\sqrt{\frac{413629}{900}}$$

Whakarūnātia.

$$y-\frac{767}{30}=\frac{\sqrt{413629}}{30} y-\frac{767}{30}=-\frac{\sqrt{413629}}{30}$$

Me tāpiri \frac{767}{30} ki ngā taha e rua o te whārite.

$$y=\frac{\sqrt{413629}+767}{30} y=\frac{767-\sqrt{413629}}{30}$$