Whakaoti i te {l}{a-b=1}{a^2+b^2=25}

$$\left\{ \begin{array} { l } { a - b = 1 } \\ { a ^ { 2 } + b ^ { 2 } = 25 } \end{array} \right.$$

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.

$$a-b=1,b^{2}+a^{2}=25$$

Whakaotia te a-b=1 mō a mā te wehe i te a i te taha mauī o te tohu ōrite.

$$a-b=1$$

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

$$a=b+1$$

Whakakapia te b+1 mō te a ki tērā atu whārite, b^{2}+a^{2}=25.

$$b^{2}+\left(b+1\right)^{2}=25$$

Pūrua b+1.

$$b^{2}+b^{2}+2b+1=25$$

Tāpiri b^{2} ki te b^{2}.

$$2b^{2}+2b+1=25$$

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

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

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

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

Pūrua 1\times 1\times 1\times 2.

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

Whakareatia -4 ki te 1+1\times 1^{2}.

$$b=\frac{-2±\sqrt{4-8\left(-24\right)}}{2\times 2}$$

Whakareatia -8 ki te -24.

$$b=\frac{-2±\sqrt{4+192}}{2\times 2}$$

Tāpiri 4 ki te 192.

$$b=\frac{-2±\sqrt{196}}{2\times 2}$$

Tuhia te pūtakerua o te 196.

$$b=\frac{-2±14}{2\times 2}$$

Whakareatia 2 ki te 1+1\times 1^{2}.

$$b=\frac{-2±14}{4}$$

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

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

Whakawehe 12 ki te 4.

$$b=3$$

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

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

Whakawehe -16 ki te 4.

$$b=-4$$

E rua ngā otinga mō b: 3 me -4. Me whakakapi 3 mō b ki te whārite a=b+1 hei kimi i te otinga hāngai mō a e pai ai ki ngā whārite e rua.

$$a=3+1$$

Tāpiri 1\times 3 ki te 1.

$$a=4$$

Me whakakapi te -4 ināianei mō te b ki te whārite a=b+1 ka whakaoti hei kimi i te otinga hāngai mō a e pai ai ki ngā whārite e rua.

$$a=-4+1$$

Tāpiri -4 ki te 1.

$$a=-3$$

Kua oti te pūnaha te whakatau.

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