Whakaoti i te {l}{x^2/9+y^2/4=1}{3x+4y=1}

$$\left. \begin{array} { l } { \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 } \\ { 3 x + 4 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{63}{29}\approx 2.172413793\text{, }y=-\frac{40}{29}\approx -1.379310345$$

$$x=-\frac{9}{5}=-1.8\text{, }y=\frac{8}{5}=1.6$$

Ngā Upane e Whakamahi ana i te Whakakapinga me te Tikanga Tātai Pūrua

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 36, arā, te tauraro pātahi he tino iti rawa te kitea o 9,4.

$$4x^{2}+9y^{2}=36$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$3x+4y=1,9y^{2}+4x^{2}=36$$

Whakaotia te 3x+4y=1 mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$3x+4y=1$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$3x=-4y+1$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{4}{3}y+\frac{1}{3}$$

Whakakapia te -\frac{4}{3}y+\frac{1}{3} mō te x ki tērā atu whārite, 9y^{2}+4x^{2}=36.

$$9y^{2}+4\left(-\frac{4}{3}y+\frac{1}{3}\right)^{2}=36$$

Pūrua -\frac{4}{3}y+\frac{1}{3}.

$$9y^{2}+4\left(\frac{16}{9}y^{2}-\frac{8}{9}y+\frac{1}{9}\right)=36$$

Whakareatia 4 ki te \frac{16}{9}y^{2}-\frac{8}{9}y+\frac{1}{9}.

$$9y^{2}+\frac{64}{9}y^{2}-\frac{32}{9}y+\frac{4}{9}=36$$

Tāpiri 9y^{2} ki te \frac{64}{9}y^{2}.

$$\frac{145}{9}y^{2}-\frac{32}{9}y+\frac{4}{9}=36$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$\frac{145}{9}y^{2}-\frac{32}{9}y-\frac{320}{9}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9+4\left(-\frac{4}{3}\right)^{2} mō a, 4\times \frac{1}{3}\left(-\frac{4}{3}\right)\times 2 mō b, me -\frac{320}{9} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{32}{9}\right)±\sqrt{\left(-\frac{32}{9}\right)^{2}-4\times \frac{145}{9}\left(-\frac{320}{9}\right)}}{2\times \frac{145}{9}}$$

Pūrua 4\times \frac{1}{3}\left(-\frac{4}{3}\right)\times 2.

$$y=\frac{-\left(-\frac{32}{9}\right)±\sqrt{\frac{1024}{81}-4\times \frac{145}{9}\left(-\frac{320}{9}\right)}}{2\times \frac{145}{9}}$$

Whakareatia -4 ki te 9+4\left(-\frac{4}{3}\right)^{2}.

$$y=\frac{-\left(-\frac{32}{9}\right)±\sqrt{\frac{1024}{81}-\frac{580}{9}\left(-\frac{320}{9}\right)}}{2\times \frac{145}{9}}$$

Whakareatia -\frac{580}{9} ki te -\frac{320}{9} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=\frac{-\left(-\frac{32}{9}\right)±\sqrt{\frac{1024+185600}{81}}}{2\times \frac{145}{9}}$$

Tāpiri \frac{1024}{81} ki te \frac{185600}{81} 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.

$$y=\frac{-\left(-\frac{32}{9}\right)±\sqrt{2304}}{2\times \frac{145}{9}}$$

Tuhia te pūtakerua o te 2304.

$$y=\frac{-\left(-\frac{32}{9}\right)±48}{2\times \frac{145}{9}}$$

Ko te tauaro o 4\times \frac{1}{3}\left(-\frac{4}{3}\right)\times 2 ko \frac{32}{9}.

$$y=\frac{\frac{32}{9}±48}{2\times \frac{145}{9}}$$

Whakareatia 2 ki te 9+4\left(-\frac{4}{3}\right)^{2}.

$$y=\frac{\frac{32}{9}±48}{\frac{290}{9}}$$

Nā, me whakaoti te whārite y=\frac{\frac{32}{9}±48}{\frac{290}{9}} ina he tāpiri te ±. Tāpiri \frac{32}{9} ki te 48.

$$y=\frac{\frac{464}{9}}{\frac{290}{9}}$$

Whakawehe \frac{464}{9} ki te \frac{290}{9} mā te whakarea \frac{464}{9} ki te tau huripoki o \frac{290}{9}.

$$y=\frac{8}{5}$$

Nā, me whakaoti te whārite y=\frac{\frac{32}{9}±48}{\frac{290}{9}} ina he tango te ±. Tango 48 mai i \frac{32}{9}.

$$y=-\frac{\frac{400}{9}}{\frac{290}{9}}$$

Whakawehe -\frac{400}{9} ki te \frac{290}{9} mā te whakarea -\frac{400}{9} ki te tau huripoki o \frac{290}{9}.

$$y=-\frac{40}{29}$$

E rua ngā otinga mō y: \frac{8}{5} me -\frac{40}{29}. Me whakakapi \frac{8}{5} mō y ki te whārite x=-\frac{4}{3}y+\frac{1}{3} hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{4}{3}\times \frac{8}{5}+\frac{1}{3}$$

Whakareatia -\frac{4}{3} ki te \frac{8}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{32}{15}+\frac{1}{3}$$

Tāpiri -\frac{4}{3}\times \frac{8}{5} ki te \frac{1}{3}.

$$x=-\frac{9}{5}$$

Me whakakapi te -\frac{40}{29} ināianei mō te y ki te whārite x=-\frac{4}{3}y+\frac{1}{3} ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{4}{3}\left(-\frac{40}{29}\right)+\frac{1}{3}$$

Whakareatia -\frac{4}{3} ki te -\frac{40}{29} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{160}{87}+\frac{1}{3}$$

Tāpiri -\frac{40}{29}\left(-\frac{4}{3}\right) ki te \frac{1}{3}.

$$x=\frac{63}{29}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{9}{5},y=\frac{8}{5}\text{ or }x=\frac{63}{29},y=-\frac{40}{29}$$