Whakaoti i te {l}{x+y=16}{x^2+y^2=64}

$$\left\{ \begin{array} { l } { x + y = 16 } \\ { x ^ { 2 } + y ^ { 2 } = 64 } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$x=8+4\sqrt{2}i\approx 8+5.656854249i\text{, }y=-4\sqrt{2}i+8\approx 8-5.656854249i$$

$$x=-4\sqrt{2}i+8\approx 8-5.656854249i\text{, }y=8+4\sqrt{2}i\approx 8+5.656854249i$$

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

Whakaotia te x+y=16 mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x+y=16$$

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

$$x=-y+16$$

Whakakapia te -y+16 mō te x ki tērā atu whārite, y^{2}+x^{2}=64.

$$y^{2}+\left(-y+16\right)^{2}=64$$

Pūrua -y+16.

$$y^{2}+y^{2}-32y+256=64$$

Tāpiri y^{2} ki te y^{2}.

$$2y^{2}-32y+256=64$$

Me tango 64 mai i ngā taha e rua o te whārite.

$$2y^{2}-32y+192=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1+1\left(-1\right)^{2} mō a, 1\times 16\left(-1\right)\times 2 mō b, me 192 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 2\times 192}}{2\times 2}$$

Pūrua 1\times 16\left(-1\right)\times 2.

$$y=\frac{-\left(-32\right)±\sqrt{1024-4\times 2\times 192}}{2\times 2}$$

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

$$y=\frac{-\left(-32\right)±\sqrt{1024-8\times 192}}{2\times 2}$$

Whakareatia -8 ki te 192.

$$y=\frac{-\left(-32\right)±\sqrt{1024-1536}}{2\times 2}$$

Tāpiri 1024 ki te -1536.

$$y=\frac{-\left(-32\right)±\sqrt{-512}}{2\times 2}$$

Tuhia te pūtakerua o te -512.

$$y=\frac{-\left(-32\right)±16\sqrt{2}i}{2\times 2}$$

Ko te tauaro o 1\times 16\left(-1\right)\times 2 ko 32.

$$y=\frac{32±16\sqrt{2}i}{2\times 2}$$

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

$$y=\frac{32±16\sqrt{2}i}{4}$$

Nā, me whakaoti te whārite y=\frac{32±16\sqrt{2}i}{4} ina he tāpiri te ±. Tāpiri 32 ki te 16i\sqrt{2}.

$$y=\frac{32+2^{\frac{9}{2}}i}{4}$$

Whakawehe 32+i\times 2^{\frac{9}{2}} ki te 4.

$$y=8+2^{\frac{5}{2}}i$$

Nā, me whakaoti te whārite y=\frac{32±16\sqrt{2}i}{4} ina he tango te ±. Tango 16i\sqrt{2} mai i 32.

$$y=\frac{-2^{\frac{9}{2}}i+32}{4}$$

Whakawehe 32-i\times 2^{\frac{9}{2}} ki te 4.

$$y=-2^{\frac{5}{2}}i+8$$

E rua ngā otinga mō y: 8+i\times 2^{\frac{5}{2}} me 8-i\times 2^{\frac{5}{2}}. Me whakakapi 8+i\times 2^{\frac{5}{2}} mō y ki te whārite x=-y+16 hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\left(8+2^{\frac{5}{2}}i\right)+16$$

Me whakakapi te 8-i\times 2^{\frac{5}{2}} ināianei mō te y ki te whārite x=-y+16 ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\left(-2^{\frac{5}{2}}i+8\right)+16$$

Kua oti te pūnaha te whakatau.

$$x=-\left(8+2^{\frac{5}{2}}i\right)+16,y=8+2^{\frac{5}{2}}i\text{ or }x=-\left(-2^{\frac{5}{2}}i+8\right)+16,y=-2^{\frac{5}{2}}i+8$$