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

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

Datrys ar gyfer x, y

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

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

Camau gan Ddefnyddio Amnewidiad a Fformiwla Cwadratig

Ystyriwch yr hafaliad cyntaf. Lluoswch ddwy ochr yr hafaliad wrth 400, lluoswm cyffredin lleiaf 16,25.

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

I ddatrys pâr o hafaliadau gan ddefnyddio amnewid, yn gyntaf datryswch un o'r hafaliadau ar gyfer un o'r newidynnau. Yna amnewidiwch y canlyniad am y newidyn hwnnw yn yr hafaliad arall.

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

Datryswch 125x-48y=481 am x drwy ynysu x ar ochr chwith yr arwydd hafal.

$$125x-48y=481$$

Tynnu -48y o ddwy ochr yr hafaliad.

$$125x=48y+481$$

Rhannu’r ddwy ochr â 125.

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

Amnewid \frac{48}{125}y+\frac{481}{125} am x yn yr hafaliad arall, -16y^{2}+25x^{2}=400.

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

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

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

Lluoswch 25 â \frac{2304}{15625}y^{2}+\frac{46176}{15625}y+\frac{231361}{15625}.

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

Adio -16y^{2} at \frac{2304}{625}y^{2}.

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

Tynnu 400 o ddwy ochr yr hafaliad.

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

Mae’r hafaliad hwn yn y ffurf safonol: ax^{2}+bx+c=0. Amnewidiwch -16+25\times \left(\frac{48}{125}\right)^{2} am a, 25\times \frac{481}{125}\times \frac{48}{125}\times 2 am b, a -\frac{18639}{625} am c yn y fformiwla gwadratig, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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

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

Lluoswch -4 â -16+25\times \left(\frac{48}{125}\right)^{2}.

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

Lluoswch \frac{30784}{625} â -\frac{18639}{625} drwy luosi'r rhifiadur â’r rhifiadur a'r enwadur â’r enwadur. Yna, dylech leihau’r ffracsiwn i’r termau isaf os yn bosibl.

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

Adio \frac{2132222976}{390625} at -\frac{573782976}{390625} drwy ddod o hyd i enwadur cyffredin ac ychwanegu’r rhifiaduron. Yna, lleihau’r ffracsiwn i’r termau isaf os yn bosibl.

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

Cymryd isradd \frac{2493504}{625}.

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

Lluoswch 2 â -16+25\times \left(\frac{48}{125}\right)^{2}.

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

Datryswch yr hafaliad y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}} pan fydd ± yn plws. Adio -\frac{46176}{625} at \frac{72\sqrt{481}}{25}.

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

Rhannwch -\frac{46176}{625}+\frac{72\sqrt{481}}{25} â -\frac{15392}{625} drwy luosi -\frac{46176}{625}+\frac{72\sqrt{481}}{25} â chilydd -\frac{15392}{625}.

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

Datryswch yr hafaliad y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}} pan fydd ± yn minws. Tynnu \frac{72\sqrt{481}}{25} o -\frac{46176}{625}.

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

Rhannwch -\frac{46176}{625}-\frac{72\sqrt{481}}{25} â -\frac{15392}{625} drwy luosi -\frac{46176}{625}-\frac{72\sqrt{481}}{25} â chilydd -\frac{15392}{625}.

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

Mae dau ateb ar gyfer y: 3-\frac{225\sqrt{481}}{1924} a 3+\frac{225\sqrt{481}}{1924}. Amnewidiwch 3-\frac{225\sqrt{481}}{1924} am y yn yr hafaliad x=\frac{48}{125}y+\frac{481}{125} i ddod o hyd i'r ateb cyfatebol ar gyfer x sy'n bodloni'r ddau hafaliad.

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

Lluoswch \frac{48}{125} â 3-\frac{225\sqrt{481}}{1924}.

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

Nawr, amnewidiwch 3+\frac{225\sqrt{481}}{1924} am y yn yr hafaliad x=\frac{48}{125}y+\frac{481}{125} a’i ddatrys i ganfod yr ateb cyfatebol ar gyfer x sy'n bodloni'r ddau hafaliad.

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

Lluoswch \frac{48}{125} â 3+\frac{225\sqrt{481}}{1924}.

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

Mae’r system wedi’i datrys nawr.

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