Datrys {l}{x+y=16}{x^2+y^2=64}

$$\left\{ \begin{array} { l } { x + y = 16 } \\ { x ^ { 2 } + y ^ { 2 } = 64 } \end{array} \right.$$

Datrys ar gyfer x, y (complex solution)

$$x=8+4\sqrt{2}i\approx 8+5.656854249i\text{, }y=-4\sqrt{2}i+8\approx 8-5.656854249i$$

$$x=-4\sqrt{2}i+8\approx 8-5.656854249i\text{, }y=8+4\sqrt{2}i\approx 8+5.656854249i$$

Camau gan Ddefnyddio Amnewidiad a Fformiwla Cwadratig

Datryswch x+y=16 am x drwy ynysu x ar ochr chwith yr arwydd hafal.

$$x+y=16$$

Tynnu y o ddwy ochr yr hafaliad.

$$x=-y+16$$

Amnewid -y+16 am x yn yr hafaliad arall, y^{2}+x^{2}=64.

$$y^{2}+\left(-y+16\right)^{2}=64$$

Sgwâr -y+16.

$$y^{2}+y^{2}-32y+256=64$$

Adio y^{2} at y^{2}.

$$2y^{2}-32y+256=64$$

Tynnu 64 o ddwy ochr yr hafaliad.

$$2y^{2}-32y+192=0$$

Mae’r hafaliad hwn yn y ffurf safonol: ax^{2}+bx+c=0. Amnewidiwch 1+1\left(-1\right)^{2} am a, 1\times 16\left(-1\right)\times 2 am b, a 192 am c yn y fformiwla gwadratig, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 2\times 192}}{2\times 2}$$

Sgwâr 1\times 16\left(-1\right)\times 2.

$$y=\frac{-\left(-32\right)±\sqrt{1024-4\times 2\times 192}}{2\times 2}$$

Lluoswch -4 â 1+1\left(-1\right)^{2}.

$$y=\frac{-\left(-32\right)±\sqrt{1024-8\times 192}}{2\times 2}$$

Lluoswch -8 â 192.

$$y=\frac{-\left(-32\right)±\sqrt{1024-1536}}{2\times 2}$$

Adio 1024 at -1536.

$$y=\frac{-\left(-32\right)±\sqrt{-512}}{2\times 2}$$

Cymryd isradd -512.

$$y=\frac{-\left(-32\right)±16\sqrt{2}i}{2\times 2}$$

Gwrthwyneb 1\times 16\left(-1\right)\times 2 yw 32.

$$y=\frac{32±16\sqrt{2}i}{2\times 2}$$

Lluoswch 2 â 1+1\left(-1\right)^{2}.

$$y=\frac{32±16\sqrt{2}i}{4}$$

Datryswch yr hafaliad y=\frac{32±16\sqrt{2}i}{4} pan fydd ± yn plws. Adio 32 at 16i\sqrt{2}.

$$y=\frac{32+2^{\frac{9}{2}}i}{4}$$

Rhannwch 32+i\times 2^{\frac{9}{2}} â 4.

$$y=8+2^{\frac{5}{2}}i$$

Datryswch yr hafaliad y=\frac{32±16\sqrt{2}i}{4} pan fydd ± yn minws. Tynnu 16i\sqrt{2} o 32.

$$y=\frac{-2^{\frac{9}{2}}i+32}{4}$$

Rhannwch 32-i\times 2^{\frac{9}{2}} â 4.

$$y=-2^{\frac{5}{2}}i+8$$

Mae dau ateb ar gyfer y: 8+i\times 2^{\frac{5}{2}} a 8-i\times 2^{\frac{5}{2}}. Amnewidiwch 8+i\times 2^{\frac{5}{2}} am y yn yr hafaliad x=-y+16 i ddod o hyd i'r ateb cyfatebol ar gyfer x sy'n bodloni'r ddau hafaliad.

$$x=-\left(8+2^{\frac{5}{2}}i\right)+16$$

Nawr, amnewidiwch 8-i\times 2^{\frac{5}{2}} am y yn yr hafaliad x=-y+16 a’i ddatrys i ganfod yr ateb cyfatebol ar gyfer x sy'n bodloni'r ddau hafaliad.

$$x=-\left(-2^{\frac{5}{2}}i+8\right)+16$$

Mae’r system wedi’i datrys nawr.

$$x=-\left(8+2^{\frac{5}{2}}i\right)+16,y=8+2^{\frac{5}{2}}i\text{ or }x=-\left(-2^{\frac{5}{2}}i+8\right)+16,y=-2^{\frac{5}{2}}i+8$$