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

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

Whakaoti mō x, y

$$x=\frac{\sqrt{7}+1}{2}\approx 1.822875656\text{, }y=\frac{1-\sqrt{7}}{2}\approx -0.822875656$$

$$x=\frac{1-\sqrt{7}}{2}\approx -0.822875656\text{, }y=\frac{\sqrt{7}+1}{2}\approx 1.822875656$$

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.

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

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

$$x+y=1$$

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

$$x=-y+1$$

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

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

Pūrua -y+1.

$$y^{2}+y^{2}-2y+1=4$$

Tāpiri y^{2} ki te y^{2}.

$$2y^{2}-2y+1=4$$

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

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

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

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

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

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

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

$$y=\frac{-\left(-2\right)±\sqrt{4-8\left(-3\right)}}{2\times 2}$$

Whakareatia -8 ki te -3.

$$y=\frac{-\left(-2\right)±\sqrt{4+24}}{2\times 2}$$

Tāpiri 4 ki te 24.

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

Tuhia te pūtakerua o te 28.

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

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

$$y=\frac{2±2\sqrt{7}}{2\times 2}$$

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

$$y=\frac{2±2\sqrt{7}}{4}$$

Nā, me whakaoti te whārite y=\frac{2±2\sqrt{7}}{4} ina he tāpiri te ±. Tāpiri 2 ki te 2\sqrt{7}.

$$y=\frac{2\sqrt{7}+2}{4}$$

Whakawehe 2+2\sqrt{7} ki te 4.

$$y=\frac{\sqrt{7}+1}{2}$$

Nā, me whakaoti te whārite y=\frac{2±2\sqrt{7}}{4} ina he tango te ±. Tango 2\sqrt{7} mai i 2.

$$y=\frac{2-2\sqrt{7}}{4}$$

Whakawehe 2-2\sqrt{7} ki te 4.

$$y=\frac{1-\sqrt{7}}{2}$$

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

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

Me whakakapi te \frac{1-\sqrt{7}}{2} ināianei mō te y ki te whārite x=-y+1 ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{\sqrt{7}+1}{2}+1,y=\frac{\sqrt{7}+1}{2}\text{ or }x=-\frac{1-\sqrt{7}}{2}+1,y=\frac{1-\sqrt{7}}{2}$$