Whakaoti i te {l}{by-ax=ab^2-a^3}{x/a+y/b=2a}

$$\left\{ \begin{array} { l } { b y - a x = a b ^ { 2 } - a ^ { 3 } } \\ { \frac { x } { a } + \frac { y } { b } = 2 a } \end{array} \right.$$

Whakaoti mō x, y

$$x=a^{2}$$

$$y=ab$$

$$b\neq 0\text{ and }a\neq 0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$by+\left(-a\right)x=ab^{2}-a^{3},\frac{1}{b}y+\frac{1}{a}x=2a$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$by+\left(-a\right)x=ab^{2}-a^{3}$$

Me tāpiri ax ki ngā taha e rua o te whārite.

$$by=ax+a\left(b-a\right)\left(a+b\right)$$

Whakawehea ngā taha e rua ki te b.

$$y=\frac{1}{b}\left(ax+a\left(b-a\right)\left(a+b\right)\right)$$

Whakareatia \frac{1}{b} ki te a\left(x-a^{2}+b^{2}\right).

$$y=\frac{a}{b}x+ab-\frac{a^{3}}{b}$$

Whakakapia te \frac{a\left(-a^{2}+x+b^{2}\right)}{b} mō te y ki tērā atu whārite, \frac{1}{b}y+\frac{1}{a}x=2a.

$$\frac{1}{b}\left(\frac{a}{b}x+ab-\frac{a^{3}}{b}\right)+\frac{1}{a}x=2a$$

Whakareatia b^{-1} ki te \frac{a\left(-a^{2}+x+b^{2}\right)}{b}.

$$\frac{a}{b^{2}}x-\frac{a^{3}}{b^{2}}+a+\frac{1}{a}x=2a$$

Tāpiri \frac{ax}{b^{2}} ki te \frac{x}{a}.

$$\left(\frac{a}{b^{2}}+\frac{1}{a}\right)x-\frac{a^{3}}{b^{2}}+a=2a$$

Me tango a-\frac{a^{3}}{b^{2}} mai i ngā taha e rua o te whārite.

$$\left(\frac{a}{b^{2}}+\frac{1}{a}\right)x=\frac{a^{3}}{b^{2}}+a$$

Whakawehea ngā taha e rua ki te \frac{a}{b^{2}}+\frac{1}{a}.

$$x=a^{2}$$

Whakaurua te a^{2} mō x ki y=\frac{a}{b}x+ab-\frac{a^{3}}{b}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{a}{b}a^{2}+ab-\frac{a^{3}}{b}$$

Whakareatia \frac{a}{b} ki te a^{2}.

$$y=\frac{a^{3}}{b}+ab-\frac{a^{3}}{b}$$

Tāpiri -\frac{a^{3}}{b}+ab ki te \frac{a^{3}}{b}.

$$y=ab$$

Kua oti te pūnaha te whakatau.

$$y=ab,x=a^{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$by+\left(-a\right)x=ab^{2}-a^{3},\frac{1}{b}y+\frac{1}{a}x=2a$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}a\left(b-a\right)\left(a+b\right)\\2a\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right).

$$inverse(\left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right))\left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right))\left(\begin{matrix}a\left(b-a\right)\left(a+b\right)\\2a\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right))\left(\begin{matrix}a\left(b-a\right)\left(a+b\right)\\2a\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}b&-a\\\frac{1}{b}&\frac{1}{a}\end{matrix}\right))\left(\begin{matrix}a\left(b-a\right)\left(a+b\right)\\2a\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{a\left(b\times \frac{1}{a}-\left(-a\right)\times \frac{1}{b}\right)}&-\frac{-a}{b\times \frac{1}{a}-\left(-a\right)\times \frac{1}{b}}\\-\frac{\frac{1}{b}}{b\times \frac{1}{a}-\left(-a\right)\times \frac{1}{b}}&\frac{b}{b\times \frac{1}{a}-\left(-a\right)\times \frac{1}{b}}\end{matrix}\right)\left(\begin{matrix}a\left(b-a\right)\left(a+b\right)\\2a\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{b}{a^{2}+b^{2}}&\frac{ba^{2}}{a^{2}+b^{2}}\\-\frac{a}{a^{2}+b^{2}}&\frac{ab^{2}}{a^{2}+b^{2}}\end{matrix}\right)\left(\begin{matrix}a\left(b-a\right)\left(a+b\right)\\2a\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{b}{a^{2}+b^{2}}a\left(b-a\right)\left(a+b\right)+\frac{ba^{2}}{a^{2}+b^{2}}\times 2a\\\left(-\frac{a}{a^{2}+b^{2}}\right)a\left(b-a\right)\left(a+b\right)+\frac{ab^{2}}{a^{2}+b^{2}}\times 2a\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}ab\\a^{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=ab,x=a^{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$by+\left(-a\right)x=ab^{2}-a^{3},\frac{1}{b}y+\frac{1}{a}x=2a$$

Kia ōrite ai a by me \frac{y}{b}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te b^{-1} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te b.

$$\frac{1}{b}by+\frac{1}{b}\left(-a\right)x=\frac{1}{b}\left(ab^{2}-a^{3}\right),b\times \frac{1}{b}y+b\times \frac{1}{a}x=b\times 2a$$

Whakarūnātia.

$$y+\left(-\frac{a}{b}\right)x=ab-\frac{a^{3}}{b},y+\frac{b}{a}x=2ab$$

Me tango y+\frac{b}{a}x=2ab mai i y+\left(-\frac{a}{b}\right)x=ab-\frac{a^{3}}{b} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\left(-\frac{a}{b}\right)x+\left(-\frac{b}{a}\right)x=ab-\frac{a^{3}}{b}-2ab$$

Tāpiri y ki te -y. Ka whakakore atu ngā kupu y me -y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-\frac{a}{b}\right)x+\left(-\frac{b}{a}\right)x=ab-\frac{a^{3}}{b}-2ab$$

Tāpiri -\frac{ax}{b} ki te -\frac{bx}{a}.

$$\left(-\frac{a}{b}-\frac{b}{a}\right)x=ab-\frac{a^{3}}{b}-2ab$$

Tāpiri -\frac{a^{3}}{b}+ab ki te -2ba.

$$\left(-\frac{a}{b}-\frac{b}{a}\right)x=-ab-\frac{a^{3}}{b}$$

Whakawehea ngā taha e rua ki te -\frac{a}{b}-\frac{b}{a}.

$$x=a^{2}$$

Whakaurua te a^{2} mō x ki \frac{1}{b}y+\frac{1}{a}x=2a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$\frac{1}{b}y+\frac{1}{a}a^{2}=2a$$

Whakareatia a^{-1} ki te a^{2}.

$$\frac{1}{b}y+a=2a$$

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

$$\frac{1}{b}y=a$$

Whakawehea ngā taha e rua ki te b^{-1}.

$$y=ab$$

Kua oti te pūnaha te whakatau.

$$y=ab,x=a^{2}$$