Whakaoti i te {l}{x^2/16-y^2/25=1}{125x-48y=481}

$$\left. \begin{array}{l}{ \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 25 } = 1 }\\{ 125 x - 48 y = 481 }\end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{108\sqrt{481}}{2405}+5\approx 4.015124774\text{, }y=-\frac{225\sqrt{481}}{1924}+3\approx 0.435220767$$

$$x=\frac{108\sqrt{481}}{2405}+5\approx 5.984875226\text{, }y=\frac{225\sqrt{481}}{1924}+3\approx 5.564779233$$

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

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 400, arā, te tauraro pātahi he tino iti rawa te kitea o 16,25.

$$25x^{2}-16y^{2}=400$$

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.

$$125x-48y=481,-16y^{2}+25x^{2}=400$$

Whakaotia te 125x-48y=481 mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$125x-48y=481$$

Me tango -48y mai i ngā taha e rua o te whārite.

$$125x=48y+481$$

Whakawehea ngā taha e rua ki te 125.

$$x=\frac{48}{125}y+\frac{481}{125}$$

Whakakapia te \frac{48}{125}y+\frac{481}{125} mō te x ki tērā atu whārite, -16y^{2}+25x^{2}=400.

$$-16y^{2}+25\left(\frac{48}{125}y+\frac{481}{125}\right)^{2}=400$$

Pūrua \frac{48}{125}y+\frac{481}{125}.

$$-16y^{2}+25\left(\frac{2304}{15625}y^{2}+\frac{46176}{15625}y+\frac{231361}{15625}\right)=400$$

Whakareatia 25 ki te \frac{2304}{15625}y^{2}+\frac{46176}{15625}y+\frac{231361}{15625}.

$$-16y^{2}+\frac{2304}{625}y^{2}+\frac{46176}{625}y+\frac{231361}{625}=400$$

Tāpiri -16y^{2} ki te \frac{2304}{625}y^{2}.

$$-\frac{7696}{625}y^{2}+\frac{46176}{625}y+\frac{231361}{625}=400$$

Me tango 400 mai i ngā taha e rua o te whārite.

$$-\frac{7696}{625}y^{2}+\frac{46176}{625}y-\frac{18639}{625}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -16+25\times \left(\frac{48}{125}\right)^{2} mō a, 25\times \frac{481}{125}\times \frac{48}{125}\times 2 mō b, me -\frac{18639}{625} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{46176}{625}±\sqrt{\left(\frac{46176}{625}\right)^{2}-4\left(-\frac{7696}{625}\right)\left(-\frac{18639}{625}\right)}}{2\left(-\frac{7696}{625}\right)}$$

Pūrua 25\times \frac{481}{125}\times \frac{48}{125}\times 2.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2132222976}{390625}-4\left(-\frac{7696}{625}\right)\left(-\frac{18639}{625}\right)}}{2\left(-\frac{7696}{625}\right)}$$

Whakareatia -4 ki te -16+25\times \left(\frac{48}{125}\right)^{2}.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2132222976}{390625}+\frac{30784}{625}\left(-\frac{18639}{625}\right)}}{2\left(-\frac{7696}{625}\right)}$$

Whakareatia \frac{30784}{625} ki te -\frac{18639}{625} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2132222976-573782976}{390625}}}{2\left(-\frac{7696}{625}\right)}$$

Tāpiri \frac{2132222976}{390625} ki te -\frac{573782976}{390625} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2493504}{625}}}{2\left(-\frac{7696}{625}\right)}$$

Tuhia te pūtakerua o te \frac{2493504}{625}.

$$y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{2\left(-\frac{7696}{625}\right)}$$

Whakareatia 2 ki te -16+25\times \left(\frac{48}{125}\right)^{2}.

$$y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}}$$

Nā, me whakaoti te whārite y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}} ina he tāpiri te ±. Tāpiri -\frac{46176}{625} ki te \frac{72\sqrt{481}}{25}.

$$y=\frac{\frac{72\sqrt{481}}{25}-\frac{46176}{625}}{-\frac{15392}{625}}$$

Whakawehe -\frac{46176}{625}+\frac{72\sqrt{481}}{25} ki te -\frac{15392}{625} mā te whakarea -\frac{46176}{625}+\frac{72\sqrt{481}}{25} ki te tau huripoki o -\frac{15392}{625}.

$$y=-\frac{225\sqrt{481}}{1924}+3$$

Nā, me whakaoti te whārite y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}} ina he tango te ±. Tango \frac{72\sqrt{481}}{25} mai i -\frac{46176}{625}.

$$y=\frac{-\frac{72\sqrt{481}}{25}-\frac{46176}{625}}{-\frac{15392}{625}}$$

Whakawehe -\frac{46176}{625}-\frac{72\sqrt{481}}{25} ki te -\frac{15392}{625} mā te whakarea -\frac{46176}{625}-\frac{72\sqrt{481}}{25} ki te tau huripoki o -\frac{15392}{625}.

$$y=\frac{225\sqrt{481}}{1924}+3$$

E rua ngā otinga mō y: 3-\frac{225\sqrt{481}}{1924} me 3+\frac{225\sqrt{481}}{1924}. Me whakakapi 3-\frac{225\sqrt{481}}{1924} mō y ki te whārite x=\frac{48}{125}y+\frac{481}{125} hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=\frac{48}{125}\left(-\frac{225\sqrt{481}}{1924}+3\right)+\frac{481}{125}$$

Whakareatia \frac{48}{125} ki te 3-\frac{225\sqrt{481}}{1924}.

$$x=\frac{48\left(-\frac{225\sqrt{481}}{1924}+3\right)+481}{125}$$

Me whakakapi te 3+\frac{225\sqrt{481}}{1924} ināianei mō te y ki te whārite x=\frac{48}{125}y+\frac{481}{125} ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=\frac{48}{125}\left(\frac{225\sqrt{481}}{1924}+3\right)+\frac{481}{125}$$

Whakareatia \frac{48}{125} ki te 3+\frac{225\sqrt{481}}{1924}.

$$x=\frac{48\left(\frac{225\sqrt{481}}{1924}+3\right)+481}{125}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{48\left(-\frac{225\sqrt{481}}{1924}+3\right)+481}{125},y=-\frac{225\sqrt{481}}{1924}+3\text{ or }x=\frac{48\left(\frac{225\sqrt{481}}{1924}+3\right)+481}{125},y=\frac{225\sqrt{481}}{1924}+3$$