Whakaoti i te {l}{x^2+y^2=6}{x-y=3}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 6 } \\ { x - y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{3-\sqrt{3}}{2}\approx 0.633974596\text{, }y=\frac{-\sqrt{3}-3}{2}\approx -2.366025404$$

$$x=\frac{\sqrt{3}+3}{2}\approx 2.366025404\text{, }y=\frac{\sqrt{3}-3}{2}\approx -0.633974596$$

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.

$$x-y=3,y^{2}+x^{2}=6$$

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

$$x-y=3$$

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

$$x=y+3$$

Whakakapia te y+3 mō te x ki tērā atu whārite, y^{2}+x^{2}=6.

$$y^{2}+\left(y+3\right)^{2}=6$$

Pūrua y+3.

$$y^{2}+y^{2}+6y+9=6$$

Tāpiri y^{2} ki te y^{2}.

$$2y^{2}+6y+9=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$2y^{2}+6y+3=0$$

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

$$y=\frac{-6±\sqrt{6^{2}-4\times 2\times 3}}{2\times 2}$$

Pūrua 1\times 3\times 1\times 2.

$$y=\frac{-6±\sqrt{36-4\times 2\times 3}}{2\times 2}$$

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

$$y=\frac{-6±\sqrt{36-8\times 3}}{2\times 2}$$

Whakareatia -8 ki te 3.

$$y=\frac{-6±\sqrt{36-24}}{2\times 2}$$

Tāpiri 36 ki te -24.

$$y=\frac{-6±\sqrt{12}}{2\times 2}$$

Tuhia te pūtakerua o te 12.

$$y=\frac{-6±2\sqrt{3}}{2\times 2}$$

Whakareatia 2 ki te 1+1\times 1^{2}.

$$y=\frac{-6±2\sqrt{3}}{4}$$

Nā, me whakaoti te whārite y=\frac{-6±2\sqrt{3}}{4} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{3}.

$$y=\frac{2\sqrt{3}-6}{4}$$

Whakawehe -6+2\sqrt{3} ki te 4.

$$y=\frac{\sqrt{3}-3}{2}$$

Nā, me whakaoti te whārite y=\frac{-6±2\sqrt{3}}{4} ina he tango te ±. Tango 2\sqrt{3} mai i -6.

$$y=\frac{-2\sqrt{3}-6}{4}$$

Whakawehe -6-2\sqrt{3} ki te 4.

$$y=\frac{-\sqrt{3}-3}{2}$$

E rua ngā otinga mō y: \frac{-3+\sqrt{3}}{2} me \frac{-3-\sqrt{3}}{2}. Me whakakapi \frac{-3+\sqrt{3}}{2} mō y ki te whārite x=y+3 hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=\frac{\sqrt{3}-3}{2}+3$$

Me whakakapi te \frac{-3-\sqrt{3}}{2} ināianei mō te y ki te whārite x=y+3 ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=\frac{-\sqrt{3}-3}{2}+3$$

Kua oti te pūnaha te whakatau.

$$x=\frac{\sqrt{3}-3}{2}+3,y=\frac{\sqrt{3}-3}{2}\text{ or }x=\frac{-\sqrt{3}-3}{2}+3,y=\frac{-\sqrt{3}-3}{2}$$