Whakaoti i te {l}{x^2+y^2=16}{x+y=sqrt{26}}

$$\left\{ \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 16 } \\ { x + y = \sqrt { 26 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{\sqrt{6}+\sqrt{26}}{2}\approx 3.774254628\text{, }y=\frac{\sqrt{26}-\sqrt{6}}{2}\approx 1.324764885$$

$$x=\frac{\sqrt{26}-\sqrt{6}}{2}\approx 1.324764885\text{, }y=\frac{\sqrt{6}+\sqrt{26}}{2}\approx 3.774254628$$

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.

$$x+y=\sqrt{26},y^{2}+x^{2}=16$$

Whakaotia te x+y=\sqrt{26} mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x+y=\sqrt{26}$$

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

$$x=-y+\sqrt{26}$$

Whakakapia te -y+\sqrt{26} mō te x ki tērā atu whārite, y^{2}+x^{2}=16.

$$y^{2}+\left(-y+\sqrt{26}\right)^{2}=16$$

Pūrua -y+\sqrt{26}.

$$y^{2}+y^{2}+\left(-2\sqrt{26}\right)y+\left(\sqrt{26}\right)^{2}=16$$

Tāpiri y^{2} ki te y^{2}.

$$2y^{2}+\left(-2\sqrt{26}\right)y+\left(\sqrt{26}\right)^{2}=16$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$2y^{2}+\left(-2\sqrt{26}\right)y+\left(\sqrt{26}\right)^{2}-16=0$$

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 2\sqrt{26} mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-2\sqrt{26}\right)±\sqrt{\left(-2\sqrt{26}\right)^{2}-4\times 2\times 10}}{2\times 2}$$

Pūrua 1\left(-1\right)\times 2\sqrt{26}.

$$y=\frac{-\left(-2\sqrt{26}\right)±\sqrt{104-4\times 2\times 10}}{2\times 2}$$

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

$$y=\frac{-\left(-2\sqrt{26}\right)±\sqrt{104-8\times 10}}{2\times 2}$$

Whakareatia -8 ki te 10.

$$y=\frac{-\left(-2\sqrt{26}\right)±\sqrt{104-80}}{2\times 2}$$

Tāpiri 104 ki te -80.

$$y=\frac{-\left(-2\sqrt{26}\right)±\sqrt{24}}{2\times 2}$$

Tuhia te pūtakerua o te 24.

$$y=\frac{-\left(-2\sqrt{26}\right)±2\sqrt{6}}{2\times 2}$$

Ko te tauaro o 1\left(-1\right)\times 2\sqrt{26} ko 2\sqrt{26}.

$$y=\frac{2\sqrt{26}±2\sqrt{6}}{2\times 2}$$

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

$$y=\frac{2\sqrt{26}±2\sqrt{6}}{4}$$

Nā, me whakaoti te whārite y=\frac{2\sqrt{26}±2\sqrt{6}}{4} ina he tāpiri te ±. Tāpiri 2\sqrt{26} ki te 2\sqrt{6}.

$$y=\frac{2\sqrt{6}+2\sqrt{26}}{4}$$

Whakawehe 2\sqrt{26}+2\sqrt{6} ki te 4.

$$y=\frac{\sqrt{6}+\sqrt{26}}{2}$$

Nā, me whakaoti te whārite y=\frac{2\sqrt{26}±2\sqrt{6}}{4} ina he tango te ±. Tango 2\sqrt{6} mai i 2\sqrt{26}.

$$y=\frac{2\sqrt{26}-2\sqrt{6}}{4}$$

Whakawehe 2\sqrt{26}-2\sqrt{6} ki te 4.

$$y=\frac{\sqrt{26}-\sqrt{6}}{2}$$

E rua ngā otinga mō y: \frac{\sqrt{26}+\sqrt{6}}{2} me \frac{\sqrt{26}-\sqrt{6}}{2}. Me whakakapi \frac{\sqrt{26}+\sqrt{6}}{2} mō y ki te whārite x=-y+\sqrt{26} hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{\sqrt{6}+\sqrt{26}}{2}+\sqrt{26}$$

Me whakakapi te \frac{\sqrt{26}-\sqrt{6}}{2} ināianei mō te y ki te whārite x=-y+\sqrt{26} ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=-\frac{\sqrt{26}-\sqrt{6}}{2}+\sqrt{26}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{\sqrt{6}+\sqrt{26}}{2}+\sqrt{26},y=\frac{\sqrt{6}+\sqrt{26}}{2}\text{ or }x=-\frac{\sqrt{26}-\sqrt{6}}{2}+\sqrt{26},y=\frac{\sqrt{26}-\sqrt{6}}{2}$$