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

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

Whakaoti mō x, y (complex solution)

$$x=\frac{\sqrt{8-a^{2}}+a}{2}\text{, }y=\frac{-\sqrt{8-a^{2}}+a}{2}$$

$$x=\frac{-\sqrt{8-a^{2}}+a}{2}\text{, }y=\frac{\sqrt{8-a^{2}}+a}{2}$$

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

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

$$x+y=a$$

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

$$x=-y+a$$

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

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

Pūrua -y+a.

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

Tāpiri y^{2} ki te y^{2}.

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

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

$$2y^{2}+\left(-2a\right)y+a^{2}-4=0$$

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

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

Pūrua 1\left(-1\right)\times 2a.

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

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

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

Whakareatia -8 ki te -4+a^{2}.

$$y=\frac{-\left(-2a\right)±\sqrt{4a^{2}+32-8a^{2}}}{2\times 2}$$

Tāpiri 4a^{2} ki te 32-8a^{2}.

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

Tuhia te pūtakerua o te 32-4a^{2}.

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

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

$$y=\frac{2a±2\sqrt{8-a^{2}}}{4}$$

Nā, me whakaoti te whārite y=\frac{2a±2\sqrt{8-a^{2}}}{4} ina he tāpiri te ±. Tāpiri 2a ki te 2\sqrt{8-a^{2}}.

$$y=\frac{2\sqrt{8-a^{2}}+2a}{4}$$

Whakawehe 2a+2\sqrt{8-a^{2}} ki te 4.

$$y=\frac{\sqrt{8-a^{2}}+a}{2}$$

Nā, me whakaoti te whārite y=\frac{2a±2\sqrt{8-a^{2}}}{4} ina he tango te ±. Tango 2\sqrt{8-a^{2}} mai i 2a.

$$y=\frac{-2\sqrt{8-a^{2}}+2a}{4}$$

Whakawehe 2a-2\sqrt{8-a^{2}} ki te 4.

$$y=\frac{-\sqrt{8-a^{2}}+a}{2}$$

E rua ngā otinga mō y: \frac{a+\sqrt{8-a^{2}}}{2} me \frac{a-\sqrt{8-a^{2}}}{2}. Me whakakapi \frac{a+\sqrt{8-a^{2}}}{2} mō y ki te whārite x=-y+a hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{\sqrt{8-a^{2}}+a}{2}+a$$

Me whakakapi te \frac{a-\sqrt{8-a^{2}}}{2} ināianei mō te y ki te whārite x=-y+a ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{-\sqrt{8-a^{2}}+a}{2}+a$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{\sqrt{8-a^{2}}+a}{2}+a,y=\frac{\sqrt{8-a^{2}}+a}{2}\text{ or }x=-\frac{-\sqrt{8-a^{2}}+a}{2}+a,y=\frac{-\sqrt{8-a^{2}}+a}{2}$$

Whakaoti mō x, y

$$x=\frac{\sqrt{8-a^{2}}+a}{2}\text{, }y=\frac{-\sqrt{8-a^{2}}+a}{2}$$

$$x=\frac{-\sqrt{8-a^{2}}+a}{2}\text{, }y=\frac{\sqrt{8-a^{2}}+a}{2}\text{, }|a|\leq 2\sqrt{2}$$

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.

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

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

$$x+y=a$$

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

$$x=-y+a$$

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

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

Pūrua -y+a.

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

Tāpiri y^{2} ki te y^{2}.

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

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

$$2y^{2}+\left(-2a\right)y+a^{2}-4=0$$

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

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

Pūrua 1\left(-1\right)\times 2a.

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

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

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

Whakareatia -8 ki te -4+a^{2}.

$$y=\frac{-\left(-2a\right)±\sqrt{4a^{2}+32-8a^{2}}}{2\times 2}$$

Tāpiri 4a^{2} ki te 32-8a^{2}.

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

Tuhia te pūtakerua o te 32-4a^{2}.

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

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

$$y=\frac{2a±2\sqrt{8-a^{2}}}{4}$$

Nā, me whakaoti te whārite y=\frac{2a±2\sqrt{8-a^{2}}}{4} ina he tāpiri te ±. Tāpiri 2a ki te 2\sqrt{8-a^{2}}.

$$y=\frac{2\sqrt{8-a^{2}}+2a}{4}$$

Whakawehe 2a+2\sqrt{8-a^{2}} ki te 4.

$$y=\frac{\sqrt{8-a^{2}}+a}{2}$$

Nā, me whakaoti te whārite y=\frac{2a±2\sqrt{8-a^{2}}}{4} ina he tango te ±. Tango 2\sqrt{8-a^{2}} mai i 2a.

$$y=\frac{-2\sqrt{8-a^{2}}+2a}{4}$$

Whakawehe 2a-2\sqrt{8-a^{2}} ki te 4.

$$y=\frac{-\sqrt{8-a^{2}}+a}{2}$$

E rua ngā otinga mō y: \frac{a+\sqrt{8-a^{2}}}{2} me \frac{a-\sqrt{8-a^{2}}}{2}. Me whakakapi \frac{a+\sqrt{8-a^{2}}}{2} mō y ki te whārite x=-y+a hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{\sqrt{8-a^{2}}+a}{2}+a$$

Me whakakapi te \frac{a-\sqrt{8-a^{2}}}{2} ināianei mō te y ki te whārite x=-y+a ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{-\sqrt{8-a^{2}}+a}{2}+a$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{\sqrt{8-a^{2}}+a}{2}+a,y=\frac{\sqrt{8-a^{2}}+a}{2}\text{ or }x=-\frac{-\sqrt{8-a^{2}}+a}{2}+a,y=\frac{-\sqrt{8-a^{2}}+a}{2}$$