Whakaoti i te {l}{9x+my+3=0}{mx+4y+2=0}

$$\left. \begin{array} { l } { 9 x + m y + 3 = 0 } \\ { m x + 4 y + 2 = 0 } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}x=-\frac{2}{m+6}\text{, }y=-\frac{3}{m+6}\text{, }&m\neq -6\\x=\frac{-2y-1}{3}\text{, }y\in \mathrm{C}\text{, }&m=6\end{matrix}\right.$$

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.

$$9x+my+3=0,mx+4y+2=0$$

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

$$9x+my+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$9x+my=-3$$

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

$$9x=\left(-m\right)y-3$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(\left(-m\right)y-3\right)$$

Whakareatia \frac{1}{9} ki te -my-3.

$$x=\left(-\frac{m}{9}\right)y-\frac{1}{3}$$

Whakakapia te -\frac{my}{9}-\frac{1}{3} mō te x ki tērā atu whārite, mx+4y+2=0.

$$m\left(\left(-\frac{m}{9}\right)y-\frac{1}{3}\right)+4y+2=0$$

Whakareatia m ki te -\frac{my}{9}-\frac{1}{3}.

$$\left(-\frac{m^{2}}{9}\right)y-\frac{m}{3}+4y+2=0$$

Tāpiri -\frac{m^{2}y}{9} ki te 4y.

$$\left(-\frac{m^{2}}{9}+4\right)y-\frac{m}{3}+2=0$$

Me tango -\frac{m}{3}+2 mai i ngā taha e rua o te whārite.

$$\left(-\frac{m^{2}}{9}+4\right)y=\frac{m}{3}-2$$

Whakawehea ngā taha e rua ki te -\frac{m^{2}}{9}+4.

$$y=-\frac{3}{m+6}$$

Whakaurua te -\frac{3}{6+m} mō y ki x=\left(-\frac{m}{9}\right)y-\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{m}{9}\right)\left(-\frac{3}{m+6}\right)-\frac{1}{3}$$

Whakareatia -\frac{m}{9} ki te -\frac{3}{6+m}.

$$x=\frac{m}{3\left(m+6\right)}-\frac{1}{3}$$

Tāpiri -\frac{1}{3} ki te \frac{m}{3\left(6+m\right)}.

$$x=-\frac{2}{m+6}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

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.

$$9x+my+3=0,mx+4y+2=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&m\\m&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9\times 4-mm}&-\frac{m}{9\times 4-mm}\\-\frac{m}{9\times 4-mm}&\frac{9}{9\times 4-mm}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}&-\frac{m}{36-m^{2}}\\-\frac{m}{36-m^{2}}&\frac{9}{36-m^{2}}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}\left(-3\right)+\left(-\frac{m}{36-m^{2}}\right)\left(-2\right)\\\left(-\frac{m}{36-m^{2}}\right)\left(-3\right)+\frac{9}{36-m^{2}}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{m+6}\\-\frac{3}{m+6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

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.

$$9x+my+3=0,mx+4y+2=0$$

Kia ōrite ai a 9x me mx, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te m me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$m\times 9x+mmy+m\times 3=0,9mx+9\times 4y+9\times 2=0$$

Whakarūnātia.

$$9mx+m^{2}y+3m=0,9mx+36y+18=0$$

Me tango 9mx+36y+18=0 mai i 9mx+m^{2}y+3m=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9mx+\left(-9m\right)x+m^{2}y-36y+3m-18=0$$

Tāpiri 9mx ki te -9mx. Ka whakakore atu ngā kupu 9mx me -9mx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$m^{2}y-36y+3m-18=0$$

Tāpiri m^{2}y ki te -36y.

$$\left(m^{2}-36\right)y+3m-18=0$$

Me tango -18+3m mai i ngā taha e rua o te whārite.

$$\left(m^{2}-36\right)y=18-3m$$

Whakawehea ngā taha e rua ki te m^{2}-36.

$$y=-\frac{3}{m+6}$$

Whakaurua te -\frac{3}{6+m} mō y ki mx+4y+2=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$mx+4\left(-\frac{3}{m+6}\right)+2=0$$

Whakareatia 4 ki te -\frac{3}{6+m}.

$$mx-\frac{12}{m+6}+2=0$$

Tāpiri -\frac{12}{6+m} ki te 2.

$$mx+\frac{2m}{m+6}=0$$

Me tango \frac{2m}{6+m} mai i ngā taha e rua o te whārite.

$$mx=-\frac{2m}{m+6}$$

Whakawehea ngā taha e rua ki te m.

$$x=-\frac{2}{m+6}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

Whakaoti mō x, y

$$\left\{\begin{matrix}x=-\frac{2}{m+6}\text{, }y=-\frac{3}{m+6}\text{, }&|m|\neq 6\\x=\frac{-2y-1}{3}\text{, }y\in \mathrm{R}\text{, }&m=6\end{matrix}\right.$$

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.

$$9x+my+3=0,mx+4y+2=0$$

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

$$9x+my+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$9x+my=-3$$

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

$$9x=\left(-m\right)y-3$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(\left(-m\right)y-3\right)$$

Whakareatia \frac{1}{9} ki te -my-3.

$$x=\left(-\frac{m}{9}\right)y-\frac{1}{3}$$

Whakakapia te -\frac{my}{9}-\frac{1}{3} mō te x ki tērā atu whārite, mx+4y+2=0.

$$m\left(\left(-\frac{m}{9}\right)y-\frac{1}{3}\right)+4y+2=0$$

Whakareatia m ki te -\frac{my}{9}-\frac{1}{3}.

$$\left(-\frac{m^{2}}{9}\right)y-\frac{m}{3}+4y+2=0$$

Tāpiri -\frac{m^{2}y}{9} ki te 4y.

$$\left(-\frac{m^{2}}{9}+4\right)y-\frac{m}{3}+2=0$$

Me tango -\frac{m}{3}+2 mai i ngā taha e rua o te whārite.

$$\left(-\frac{m^{2}}{9}+4\right)y=\frac{m}{3}-2$$

Whakawehea ngā taha e rua ki te -\frac{m^{2}}{9}+4.

$$y=-\frac{3}{m+6}$$

Whakaurua te -\frac{3}{6+m} mō y ki x=\left(-\frac{m}{9}\right)y-\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{m}{9}\right)\left(-\frac{3}{m+6}\right)-\frac{1}{3}$$

Whakareatia -\frac{m}{9} ki te -\frac{3}{6+m}.

$$x=\frac{m}{3\left(m+6\right)}-\frac{1}{3}$$

Tāpiri -\frac{1}{3} ki te \frac{m}{3\left(6+m\right)}.

$$x=-\frac{2}{m+6}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

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.

$$9x+my+3=0,mx+4y+2=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&m\\m&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9\times 4-mm}&-\frac{m}{9\times 4-mm}\\-\frac{m}{9\times 4-mm}&\frac{9}{9\times 4-mm}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}&-\frac{m}{36-m^{2}}\\-\frac{m}{36-m^{2}}&\frac{9}{36-m^{2}}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}\left(-3\right)+\left(-\frac{m}{36-m^{2}}\right)\left(-2\right)\\\left(-\frac{m}{36-m^{2}}\right)\left(-3\right)+\frac{9}{36-m^{2}}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{m+6}\\-\frac{3}{m+6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

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.

$$9x+my+3=0,mx+4y+2=0$$

Kia ōrite ai a 9x me mx, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te m me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$m\times 9x+mmy+m\times 3=0,9mx+9\times 4y+9\times 2=0$$

Whakarūnātia.

$$9mx+m^{2}y+3m=0,9mx+36y+18=0$$

Me tango 9mx+36y+18=0 mai i 9mx+m^{2}y+3m=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9mx+\left(-9m\right)x+m^{2}y-36y+3m-18=0$$

Tāpiri 9mx ki te -9mx. Ka whakakore atu ngā kupu 9mx me -9mx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$m^{2}y-36y+3m-18=0$$

Tāpiri m^{2}y ki te -36y.

$$\left(m^{2}-36\right)y+3m-18=0$$

Me tango -18+3m mai i ngā taha e rua o te whārite.

$$\left(m^{2}-36\right)y=18-3m$$

Whakawehea ngā taha e rua ki te m^{2}-36.

$$y=-\frac{3}{m+6}$$

Whakaurua te -\frac{3}{6+m} mō y ki mx+4y+2=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$mx+4\left(-\frac{3}{m+6}\right)+2=0$$

Whakareatia 4 ki te -\frac{3}{6+m}.

$$mx-\frac{12}{m+6}+2=0$$

Tāpiri -\frac{12}{6+m} ki te 2.

$$mx+\frac{2m}{m+6}=0$$

Me tango \frac{2m}{6+m} mai i ngā taha e rua o te whārite.

$$mx=-\frac{2m}{m+6}$$

Whakawehea ngā taha e rua ki te m.

$$x=-\frac{2}{m+6}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$