Réitigh {l}{x+y=16}{x^2+y^2=64}

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

Réitigh do x,y. (complex solution)

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

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

Céimeanna ag Baint Úsáid as Ionadú agus an bhFoirme Chearnach

Réitigh x+y=16 do x trí x ar an taobh clé den chomhartha ‘Cothrom le’ a aonrú.

$$x+y=16$$

Bain y ón dá thaobh den chothromóid.

$$x=-y+16$$

Cuir x in aonad -y+16 sa chothromóid eile, y^{2}+x^{2}=64.

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

Cearnóg -y+16.

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

Suimigh y^{2} le y^{2}?

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

Bain 64 ón dá thaobh den chothromóid.

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

Tá an chothromóid seo i bhfoirm chaighdeánach: ax^{2}+bx+c=0. Cuir 1+1\left(-1\right)^{2} in ionad a, 1\times 16\left(-1\right)\times 2 in ionad b, agus 192 in ionad c san fhoirmle chearnach, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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

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

Méadaigh -4 faoi 1+1\left(-1\right)^{2}.

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

Méadaigh -8 faoi 192.

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

Suimigh 1024 le -1536?

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

Tóg fréamh chearnach -512.

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

Tá 32 urchomhairleach le 1\times 16\left(-1\right)\times 2.

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

Méadaigh 2 faoi 1+1\left(-1\right)^{2}.

$$y=\frac{32±16\sqrt{2}i}{4}$$

Réitigh an chothromóid y=\frac{32±16\sqrt{2}i}{4} nuair is ionann ± agus plus. Suimigh 32 le 16i\sqrt{2}?

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

Roinn 32+i\times 2^{\frac{9}{2}} faoi 4.

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

Réitigh an chothromóid y=\frac{32±16\sqrt{2}i}{4} nuair is ionann ± agus míneas. Dealaigh 16i\sqrt{2} ó 32.

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

Roinn 32-i\times 2^{\frac{9}{2}} faoi 4.

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

Tá dhá réiteach ann do y: 8+i\times 2^{\frac{5}{2}} agus 8-i\times 2^{\frac{5}{2}}. Cuir y in aonad 8+i\times 2^{\frac{5}{2}} sa chothromóid eile x=-y+16 chun an réiteach comhfhreagrach do x a shásaíonn an dá chothromóid a fháil.

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

Ansin cuir y in aonad 8-i\times 2^{\frac{5}{2}} sa chothromóid eile x=-y+16 agus faigh réiteach chun an réiteach comhfhreagrach do x a shásaíonn an dá chothromóid a fháil.

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

Tá an córas réitithe anois.

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