Whakaoti i te {l}{y=kx+b}{x^2/4+y^2=1}

$$\left\{ \begin{array} { l } { y = k x + b } \\ { \frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\text{, }y=\frac{-2k\sqrt{1+4k^{2}-b^{2}}+b}{4k^{2}+1}$$

$$x=\frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\text{, }y=\frac{2k\sqrt{1+4k^{2}-b^{2}}+b}{4k^{2}+1}\text{, }|k|\geq \frac{\sqrt{b^{2}-1}}{2}\text{ or }|b|<1$$

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

Whakaarohia te whārite tuatahi. Tangohia te kx mai i ngā taha e rua.

$$y-kx=b$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 4.

$$x^{2}+4y^{2}=4$$

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.

$$y+\left(-k\right)x=b,x^{2}+4y^{2}=4$$

Whakaotia te y+\left(-k\right)x=b mō y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$y+\left(-k\right)x=b$$

Me tango \left(-k\right)x mai i ngā taha e rua o te whārite.

$$y=kx+b$$

Whakakapia te kx+b mō te y ki tērā atu whārite, x^{2}+4y^{2}=4.

$$x^{2}+4\left(kx+b\right)^{2}=4$$

Pūrua kx+b.

$$x^{2}+4\left(k^{2}x^{2}+2bkx+b^{2}\right)=4$$

Whakareatia 4 ki te k^{2}x^{2}+2bkx+b^{2}.

$$x^{2}+4k^{2}x^{2}+8bkx+4b^{2}=4$$

Tāpiri x^{2} ki te 4k^{2}x^{2}.

$$\left(4k^{2}+1\right)x^{2}+8bkx+4b^{2}=4$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$\left(4k^{2}+1\right)x^{2}+8bkx+4b^{2}-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1+4k^{2} mō a, 4\times 2kb mō b, me -4+4b^{2} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8bk±\sqrt{\left(8bk\right)^{2}-4\left(4k^{2}+1\right)\left(4b^{2}-4\right)}}{2\left(4k^{2}+1\right)}$$

Pūrua 4\times 2kb.

$$x=\frac{-8bk±\sqrt{64b^{2}k^{2}-4\left(4k^{2}+1\right)\left(4b^{2}-4\right)}}{2\left(4k^{2}+1\right)}$$

Whakareatia -4 ki te 1+4k^{2}.

$$x=\frac{-8bk±\sqrt{64b^{2}k^{2}+\left(-16k^{2}-4\right)\left(4b^{2}-4\right)}}{2\left(4k^{2}+1\right)}$$

Whakareatia -4-16k^{2} ki te -4+4b^{2}.

$$x=\frac{-8bk±\sqrt{64b^{2}k^{2}-16\left(b^{2}-1\right)\left(4k^{2}+1\right)}}{2\left(4k^{2}+1\right)}$$

Tāpiri 64k^{2}b^{2} ki te -16\left(1+4k^{2}\right)\left(b^{2}-1\right).

$$x=\frac{-8bk±\sqrt{16+64k^{2}-16b^{2}}}{2\left(4k^{2}+1\right)}$$

Tuhia te pūtakerua o te -16b^{2}+64k^{2}+16.

$$x=\frac{-8bk±4\sqrt{1+4k^{2}-b^{2}}}{2\left(4k^{2}+1\right)}$$

Whakareatia 2 ki te 1+4k^{2}.

$$x=\frac{-8bk±4\sqrt{1+4k^{2}-b^{2}}}{8k^{2}+2}$$

Nā, me whakaoti te whārite x=\frac{-8bk±4\sqrt{1+4k^{2}-b^{2}}}{8k^{2}+2} ina he tāpiri te ±. Tāpiri -8kb ki te 4\sqrt{-b^{2}+4k^{2}+1}.

$$x=\frac{-8bk+4\sqrt{1+4k^{2}-b^{2}}}{8k^{2}+2}$$

Whakawehe -8bk+4\sqrt{-b^{2}+4k^{2}+1} ki te 2+8k^{2}.

$$x=\frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}$$

Nā, me whakaoti te whārite x=\frac{-8bk±4\sqrt{1+4k^{2}-b^{2}}}{8k^{2}+2} ina he tango te ±. Tango 4\sqrt{-b^{2}+4k^{2}+1} mai i -8kb.

$$x=\frac{-8bk-4\sqrt{1+4k^{2}-b^{2}}}{8k^{2}+2}$$

Whakawehe -8kb-4\sqrt{-b^{2}+4k^{2}+1} ki te 2+8k^{2}.

$$x=-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}$$

E rua ngā otinga mō x: \frac{2\left(-2bk+\sqrt{-b^{2}+4k^{2}+1}\right)}{1+4k^{2}} me -\frac{2\left(2bk+\sqrt{-b^{2}+4k^{2}+1}\right)}{1+4k^{2}}. Me whakakapi \frac{2\left(-2bk+\sqrt{-b^{2}+4k^{2}+1}\right)}{1+4k^{2}} mō x ki te whārite y=kx+b hei kimi i te otinga hāngai mō y e pai ai ki ngā whārite e rua.

$$y=k\times \frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}+b$$

Whakareatia k ki te \frac{2\left(-2bk+\sqrt{-b^{2}+4k^{2}+1}\right)}{1+4k^{2}}.

$$y=\frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}k+b$$

Me whakakapi te -\frac{2\left(2bk+\sqrt{-b^{2}+4k^{2}+1}\right)}{1+4k^{2}} ināianei mō te x ki te whārite y=kx+b ka whakaoti hei kimi i te otinga hāngai mō y e pai ai ki ngā whārite e rua.

$$y=k\left(-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\right)+b$$

Whakareatia k ki te -\frac{2\left(2bk+\sqrt{-b^{2}+4k^{2}+1}\right)}{1+4k^{2}}.

$$y=\left(-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\right)k+b$$

Kua oti te pūnaha te whakatau.

$$y=\frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}k+b,x=\frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\text{ or }y=\left(-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\right)k+b,x=-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}x=-\frac{2\left(2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\text{, }y=\frac{-2k\sqrt{1+4k^{2}-b^{2}}+b}{4k^{2}+1}\text{; }x=\frac{2\left(-2bk+\sqrt{1+4k^{2}-b^{2}}\right)}{4k^{2}+1}\text{, }y=\frac{2k\sqrt{1+4k^{2}-b^{2}}+b}{4k^{2}+1}\text{, }&k\neq -\frac{1}{2}i\text{ and }k\neq \frac{1}{2}i\\x=-\frac{b^{2}-1}{2bk}\text{, }y=\frac{b^{2}+1}{2b}\text{, }&b\neq 0\text{ and }\left(k=-\frac{1}{2}i\text{ or }k=\frac{1}{2}i\right)\end{matrix}\right.$$