Whakaoti i te {l}{x^2+y^2=9}{x+y=2}

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

Whakaoti mō x, y

$$x=\frac{\sqrt{14}}{2}+1\approx 2.870828693\text{, }y=-\frac{\sqrt{14}}{2}+1\approx -0.870828693$$

$$x=-\frac{\sqrt{14}}{2}+1\approx -0.870828693\text{, }y=\frac{\sqrt{14}}{2}+1\approx 2.870828693$$

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.

$$x+y=2,y^{2}+x^{2}=9$$

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

$$x+y=2$$

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

$$x=-y+2$$

Whakakapia te -y+2 mō te x ki tērā atu whārite, y^{2}+x^{2}=9.

$$y^{2}+\left(-y+2\right)^{2}=9$$

Pūrua -y+2.

$$y^{2}+y^{2}-4y+4=9$$

Tāpiri y^{2} ki te y^{2}.

$$2y^{2}-4y+4=9$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$2y^{2}-4y-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1+1\left(-1\right)^{2} mō a, 1\times 2\left(-1\right)\times 2 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

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

$$y=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-5\right)}}{2\times 2}$$

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

$$y=\frac{-\left(-4\right)±\sqrt{16-8\left(-5\right)}}{2\times 2}$$

Whakareatia -8 ki te -5.

$$y=\frac{-\left(-4\right)±\sqrt{16+40}}{2\times 2}$$

Tāpiri 16 ki te 40.

$$y=\frac{-\left(-4\right)±\sqrt{56}}{2\times 2}$$

Tuhia te pūtakerua o te 56.

$$y=\frac{-\left(-4\right)±2\sqrt{14}}{2\times 2}$$

Ko te tauaro o 1\times 2\left(-1\right)\times 2 ko 4.

$$y=\frac{4±2\sqrt{14}}{2\times 2}$$

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

$$y=\frac{4±2\sqrt{14}}{4}$$

Nā, me whakaoti te whārite y=\frac{4±2\sqrt{14}}{4} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{14}.

$$y=\frac{2\sqrt{14}+4}{4}$$

Whakawehe 4+2\sqrt{14} ki te 4.

$$y=\frac{\sqrt{14}}{2}+1$$

Nā, me whakaoti te whārite y=\frac{4±2\sqrt{14}}{4} ina he tango te ±. Tango 2\sqrt{14} mai i 4.

$$y=\frac{4-2\sqrt{14}}{4}$$

Whakawehe 4-2\sqrt{14} ki te 4.

$$y=-\frac{\sqrt{14}}{2}+1$$

E rua ngā otinga mō y: 1+\frac{\sqrt{14}}{2} me 1-\frac{\sqrt{14}}{2}. Me whakakapi 1+\frac{\sqrt{14}}{2} mō y ki te whārite x=-y+2 hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\left(\frac{\sqrt{14}}{2}+1\right)+2$$

Me whakakapi te 1-\frac{\sqrt{14}}{2} ināianei mō te y ki te whārite x=-y+2 ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\left(-\frac{\sqrt{14}}{2}+1\right)+2$$

Kua oti te pūnaha te whakatau.

$$x=-\left(\frac{\sqrt{14}}{2}+1\right)+2,y=\frac{\sqrt{14}}{2}+1\text{ or }x=-\left(-\frac{\sqrt{14}}{2}+1\right)+2,y=-\frac{\sqrt{14}}{2}+1$$