Whakaoti i te {l}{2x^2-y^2=17}{x+2y=1}

$$\left. \begin{array}{l}{ 2 x ^ { 2 } - y ^ { 2 } = 17 }\\{ x + 2 y = 1 }\end{array} \right.$$

Whakaoti mō x, y

$$x=3\text{, }y=-1$$

$$x=-\frac{23}{7}\approx -3.285714286\text{, }y=\frac{15}{7}\approx 2.142857143$$

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

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.

$$x+2y=1,-y^{2}+2x^{2}=17$$

Whakaotia te x+2y=1 mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x+2y=1$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y+1$$

Whakakapia te -2y+1 mō te x ki tērā atu whārite, -y^{2}+2x^{2}=17.

$$-y^{2}+2\left(-2y+1\right)^{2}=17$$

Pūrua -2y+1.

$$-y^{2}+2\left(4y^{2}-4y+1\right)=17$$

Whakareatia 2 ki te 4y^{2}-4y+1.

$$-y^{2}+8y^{2}-8y+2=17$$

Tāpiri -y^{2} ki te 8y^{2}.

$$7y^{2}-8y+2=17$$

Me tango 17 mai i ngā taha e rua o te whārite.

$$7y^{2}-8y-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1+2\left(-2\right)^{2} mō a, 2\times 1\left(-2\right)\times 2 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 7\left(-15\right)}}{2\times 7}$$

Pūrua 2\times 1\left(-2\right)\times 2.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\times 7\left(-15\right)}}{2\times 7}$$

Whakareatia -4 ki te -1+2\left(-2\right)^{2}.

$$y=\frac{-\left(-8\right)±\sqrt{64-28\left(-15\right)}}{2\times 7}$$

Whakareatia -28 ki te -15.

$$y=\frac{-\left(-8\right)±\sqrt{64+420}}{2\times 7}$$

Tāpiri 64 ki te 420.

$$y=\frac{-\left(-8\right)±\sqrt{484}}{2\times 7}$$

Tuhia te pūtakerua o te 484.

$$y=\frac{-\left(-8\right)±22}{2\times 7}$$

Ko te tauaro o 2\times 1\left(-2\right)\times 2 ko 8.

$$y=\frac{8±22}{2\times 7}$$

Whakareatia 2 ki te -1+2\left(-2\right)^{2}.

$$y=\frac{8±22}{14}$$

Nā, me whakaoti te whārite y=\frac{8±22}{14} ina he tāpiri te ±. Tāpiri 8 ki te 22.

$$y=\frac{30}{14}$$

Whakahekea te hautanga \frac{30}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=\frac{15}{7}$$

Nā, me whakaoti te whārite y=\frac{8±22}{14} ina he tango te ±. Tango 22 mai i 8.

$$y=-\frac{14}{14}$$

Whakawehe -14 ki te 14.

$$y=-1$$

E rua ngā otinga mō y: \frac{15}{7} me -1. Me whakakapi \frac{15}{7} mō y ki te whārite x=-2y+1 hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-2\times \frac{15}{7}+1$$

Whakareatia -2 ki te \frac{15}{7}.

$$x=-\frac{30}{7}+1$$

Tāpiri -2\times \frac{15}{7} ki te 1.

$$x=-\frac{23}{7}$$

Me whakakapi te -1 ināianei mō te y ki te whārite x=-2y+1 ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-2\left(-1\right)+1$$

Whakareatia -2 ki te -1.

$$x=2+1$$

Tāpiri -2\left(-1\right) ki te 1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{23}{7},y=\frac{15}{7}\text{ or }x=3,y=-1$$