Whakaoti i te {l}{a+b=7}{a^2+b^2=25}

$$\left\{ \begin{array} { l } { a + b = 7 } \\ { a ^ { 2 } + b ^ { 2 } = 25 } \end{array} \right.$$

Whakaoti mō a, b

$$a=4\text{, }b=3$$

$$a=3\text{, }b=4$$

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.

$$a+b=7,b^{2}+a^{2}=25$$

Whakaotia te a+b=7 mō a mā te wehe i te a i te taha mauī o te tohu ōrite.

$$a+b=7$$

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

$$a=-b+7$$

Whakakapia te -b+7 mō te a ki tērā atu whārite, b^{2}+a^{2}=25.

$$b^{2}+\left(-b+7\right)^{2}=25$$

Pūrua -b+7.

$$b^{2}+b^{2}-14b+49=25$$

Tāpiri b^{2} ki te b^{2}.

$$2b^{2}-14b+49=25$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$2b^{2}-14b+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1+1\left(-1\right)^{2} mō a, 1\times 7\left(-1\right)\times 2 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 24}}{2\times 2}$$

Pūrua 1\times 7\left(-1\right)\times 2.

$$b=\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 24}}{2\times 2}$$

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

$$b=\frac{-\left(-14\right)±\sqrt{196-8\times 24}}{2\times 2}$$

Whakareatia -8 ki te 24.

$$b=\frac{-\left(-14\right)±\sqrt{196-192}}{2\times 2}$$

Tāpiri 196 ki te -192.

$$b=\frac{-\left(-14\right)±\sqrt{4}}{2\times 2}$$

Tuhia te pūtakerua o te 4.

$$b=\frac{-\left(-14\right)±2}{2\times 2}$$

Ko te tauaro o 1\times 7\left(-1\right)\times 2 ko 14.

$$b=\frac{14±2}{2\times 2}$$

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

$$b=\frac{14±2}{4}$$

Nā, me whakaoti te whārite b=\frac{14±2}{4} ina he tāpiri te ±. Tāpiri 14 ki te 2.

$$b=\frac{16}{4}$$

Whakawehe 16 ki te 4.

$$b=4$$

Nā, me whakaoti te whārite b=\frac{14±2}{4} ina he tango te ±. Tango 2 mai i 14.

$$b=\frac{12}{4}$$

Whakawehe 12 ki te 4.

$$b=3$$

E rua ngā otinga mō b: 4 me 3. Me whakakapi 4 mō b ki te whārite a=-b+7 hei kimi i te otinga hāngai mō a e pai ai ki ngā whārite e rua.

$$a=-4+7$$

Tāpiri -4 ki te 7.

$$a=3$$

Me whakakapi te 3 ināianei mō te b ki te whārite a=-b+7 ka whakaoti hei kimi i te otinga hāngai mō a e pai ai ki ngā whārite e rua.

$$a=-3+7$$

Tāpiri -3 ki te 7.

$$a=4$$

Kua oti te pūnaha te whakatau.

$$a=3,b=4\text{ or }a=4,b=3$$