Whakaoti i te {l}{a^2+b^2=100}{a+b=20}

$$\left\{ \begin{array} { l } { a ^ { 2 } + b ^ { 2 } = 100 } \\ { a + b = 20 } \end{array} \right.$$

Whakaoti mō a, b

$$a=10+5\sqrt{2}i\approx 10+7.071067812i\text{, }b=-5\sqrt{2}i+10\approx 10-7.071067812i$$

$$a=-5\sqrt{2}i+10\approx 10-7.071067812i\text{, }b=10+5\sqrt{2}i\approx 10+7.071067812i$$

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

Whakaotia te a+b=20 mō a mā te wehe i te a i te taha mauī o te tohu ōrite.

$$a+b=20$$

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

$$a=-b+20$$

Whakakapia te -b+20 mō te a ki tērā atu whārite, b^{2}+a^{2}=100.

$$b^{2}+\left(-b+20\right)^{2}=100$$

Pūrua -b+20.

$$b^{2}+b^{2}-40b+400=100$$

Tāpiri b^{2} ki te b^{2}.

$$2b^{2}-40b+400=100$$

Me tango 100 mai i ngā taha e rua o te whārite.

$$2b^{2}-40b+300=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1+1\left(-1\right)^{2} mō a, 1\times 20\left(-1\right)\times 2 mō b, me 300 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 2\times 300}}{2\times 2}$$

Pūrua 1\times 20\left(-1\right)\times 2.

$$b=\frac{-\left(-40\right)±\sqrt{1600-4\times 2\times 300}}{2\times 2}$$

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

$$b=\frac{-\left(-40\right)±\sqrt{1600-8\times 300}}{2\times 2}$$

Whakareatia -8 ki te 300.

$$b=\frac{-\left(-40\right)±\sqrt{1600-2400}}{2\times 2}$$

Tāpiri 1600 ki te -2400.

$$b=\frac{-\left(-40\right)±\sqrt{-800}}{2\times 2}$$

Tuhia te pūtakerua o te -800.

$$b=\frac{-\left(-40\right)±20\sqrt{2}i}{2\times 2}$$

Ko te tauaro o 1\times 20\left(-1\right)\times 2 ko 40.

$$b=\frac{40±20\sqrt{2}i}{2\times 2}$$

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

$$b=\frac{40±20\sqrt{2}i}{4}$$

Nā, me whakaoti te whārite b=\frac{40±20\sqrt{2}i}{4} ina he tāpiri te ±. Tāpiri 40 ki te 20i\sqrt{2}.

$$b=\frac{40+20\sqrt{2}i}{4}$$

Whakawehe 40+20i\sqrt{2} ki te 4.

$$b=10+5\sqrt{2}i$$

Nā, me whakaoti te whārite b=\frac{40±20\sqrt{2}i}{4} ina he tango te ±. Tango 20i\sqrt{2} mai i 40.

$$b=\frac{-20\sqrt{2}i+40}{4}$$

Whakawehe 40-20i\sqrt{2} ki te 4.

$$b=-5\sqrt{2}i+10$$

E rua ngā otinga mō b: 10+5i\sqrt{2} me 10-5i\sqrt{2}. Me whakakapi 10+5i\sqrt{2} mō b ki te whārite a=-b+20 hei kimi i te otinga hāngai mō a e pai ai ki ngā whārite e rua.

$$a=-\left(10+5\sqrt{2}i\right)+20$$

Me whakakapi te 10-5i\sqrt{2} ināianei mō te b ki te whārite a=-b+20 ka whakaoti hei kimi i te otinga hāngai mō a e pai ai ki ngā whārite e rua.

$$a=-\left(-5\sqrt{2}i+10\right)+20$$

Kua oti te pūnaha te whakatau.

$$a=-\left(10+5\sqrt{2}i\right)+20,b=10+5\sqrt{2}i\text{ or }a=-\left(-5\sqrt{2}i+10\right)+20,b=-5\sqrt{2}i+10$$