Réitigh {l}{y=3x+8}{x^2+y^2=4}

$$\left\{ \begin{array} { l } { y = 3 x + 8 } \\ { x ^ { 2 } + y ^ { 2 } = 4 } \end{array} \right.$$

Réitigh do y,x. (complex solution)

$$x=\frac{-\sqrt{6}i-12}{5}\approx -2.4-0.489897949i\text{, }y=\frac{-3\sqrt{6}i+4}{5}\approx 0.8-1.469693846i$$

$$x=\frac{-12+\sqrt{6}i}{5}\approx -2.4+0.489897949i\text{, }y=\frac{4+3\sqrt{6}i}{5}\approx 0.8+1.469693846i$$

Céimeanna ag Baint Úsáid as Ionadú agus an bhFoirme Chearnach

Cuir an chéad cothromóid san áireamh. Bain 3x ón dá thaobh.

$$y-3x=8$$

Bain -3x ón dá thaobh den chothromóid.

$$y=3x+8$$

Cuir y in aonad 3x+8 sa chothromóid eile, x^{2}+y^{2}=4.

$$x^{2}+\left(3x+8\right)^{2}=4$$

Cearnóg 3x+8.

$$x^{2}+9x^{2}+48x+64=4$$

Suimigh x^{2} le 9x^{2}?

$$10x^{2}+48x+64=4$$

Bain 4 ón dá thaobh den chothromóid.

$$10x^{2}+48x+60=0$$

Tá an chothromóid seo i bhfoirm chaighdeánach: ax^{2}+bx+c=0. Cuir 1+1\times 3^{2} in ionad a, 1\times 8\times 2\times 3 in ionad b, agus 60 in ionad c san fhoirmle chearnach, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 10\times 60}}{2\times 10}$$

Cearnóg 1\times 8\times 2\times 3.

$$x=\frac{-48±\sqrt{2304-4\times 10\times 60}}{2\times 10}$$

Méadaigh -4 faoi 1+1\times 3^{2}.

$$x=\frac{-48±\sqrt{2304-40\times 60}}{2\times 10}$$

Méadaigh -40 faoi 60.

$$x=\frac{-48±\sqrt{2304-2400}}{2\times 10}$$

Suimigh 2304 le -2400?

$$x=\frac{-48±\sqrt{-96}}{2\times 10}$$

Tóg fréamh chearnach -96.

$$x=\frac{-48±4\sqrt{6}i}{2\times 10}$$

Méadaigh 2 faoi 1+1\times 3^{2}.

$$x=\frac{-48±4\sqrt{6}i}{20}$$

Réitigh an chothromóid x=\frac{-48±4\sqrt{6}i}{20} nuair is ionann ± agus plus. Suimigh -48 le 4i\sqrt{6}?

$$x=\frac{-48+4\sqrt{6}i}{20}$$

Roinn -48+4i\sqrt{6} faoi 20.

$$x=\frac{-12+\sqrt{6}i}{5}$$

Réitigh an chothromóid x=\frac{-48±4\sqrt{6}i}{20} nuair is ionann ± agus míneas. Dealaigh 4i\sqrt{6} ó -48.

$$x=\frac{-4\sqrt{6}i-48}{20}$$

Roinn -48-4i\sqrt{6} faoi 20.

$$x=\frac{-\sqrt{6}i-12}{5}$$

Tá dhá réiteach ann do x: \frac{-12+i\sqrt{6}}{5} agus \frac{-12-i\sqrt{6}}{5}. Cuir x in aonad \frac{-12+i\sqrt{6}}{5} sa chothromóid eile y=3x+8 chun an réiteach comhfhreagrach do y a shásaíonn an dá chothromóid a fháil.

$$y=3\times \frac{-12+\sqrt{6}i}{5}+8$$

Ansin cuir x in aonad \frac{-12-i\sqrt{6}}{5} sa chothromóid eile y=3x+8 agus faigh réiteach chun an réiteach comhfhreagrach do y a shásaíonn an dá chothromóid a fháil.

$$y=3\times \frac{-\sqrt{6}i-12}{5}+8$$

Tá an córas réitithe anois.

$$y=3\times \frac{-12+\sqrt{6}i}{5}+8,x=\frac{-12+\sqrt{6}i}{5}\text{ or }y=3\times \frac{-\sqrt{6}i-12}{5}+8,x=\frac{-\sqrt{6}i-12}{5}$$