Whakaoti i te {l}{2ax+by=14}{-2x+9y=-19}

$$\left\{ \begin{array} { l } { 2 a x + b y = 14 } \\ { - 2 x + 9 y = - 19 } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}x=\frac{19b+126}{2\left(9a+b\right)}\text{, }y=-\frac{19a-14}{9a+b}\text{, }&a\neq -\frac{b}{9}\\x=\frac{9y+19}{2}\text{, }y\in \mathrm{C}\text{, }&b=-\frac{126}{19}\text{ and }a=\frac{14}{19}\end{matrix}\right.$$

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.

$$2ax+by=14,-2x+9y=-19$$

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

$$2ax+by=14$$

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

$$2ax=\left(-b\right)y+14$$

Whakawehea ngā taha e rua ki te 2a.

$$x=\frac{1}{2a}\left(\left(-b\right)y+14\right)$$

Whakareatia \frac{1}{2a} ki te -by+14.

$$x=\left(-\frac{b}{2a}\right)y+\frac{7}{a}$$

Whakakapia te \frac{-by+14}{2a} mō te x ki tērā atu whārite, -2x+9y=-19.

$$-2\left(\left(-\frac{b}{2a}\right)y+\frac{7}{a}\right)+9y=-19$$

Whakareatia -2 ki te \frac{-by+14}{2a}.

$$\frac{b}{a}y-\frac{14}{a}+9y=-19$$

Tāpiri \frac{by}{a} ki te 9y.

$$\left(\frac{b}{a}+9\right)y-\frac{14}{a}=-19$$

Me tāpiri \frac{14}{a} ki ngā taha e rua o te whārite.

$$\left(\frac{b}{a}+9\right)y=-19+\frac{14}{a}$$

Whakawehea ngā taha e rua ki te 9+\frac{b}{a}.

$$y=\frac{14-19a}{9a+b}$$

Whakaurua te \frac{14-19a}{9a+b} mō y ki x=\left(-\frac{b}{2a}\right)y+\frac{7}{a}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{b}{2a}\right)\times \frac{14-19a}{9a+b}+\frac{7}{a}$$

Whakareatia -\frac{b}{2a} ki te \frac{14-19a}{9a+b}.

$$x=-\frac{b\left(14-19a\right)}{2a\left(9a+b\right)}+\frac{7}{a}$$

Tāpiri \frac{7}{a} ki te -\frac{b\left(14-19a\right)}{2a\left(9a+b\right)}.

$$x=\frac{19b+126}{2\left(9a+b\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19b+126}{2\left(9a+b\right)},y=\frac{14-19a}{9a+b}$$

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.

$$2ax+by=14,-2x+9y=-19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2a&b\\-2&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}14\\-19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}14\\-19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}14\\-19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2a\times 9-b\left(-2\right)}&-\frac{b}{2a\times 9-b\left(-2\right)}\\-\frac{-2}{2a\times 9-b\left(-2\right)}&\frac{2a}{2a\times 9-b\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}14\\-19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\left(9a+b\right)}&-\frac{b}{2\left(9a+b\right)}\\\frac{1}{9a+b}&\frac{a}{9a+b}\end{matrix}\right)\left(\begin{matrix}14\\-19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\left(9a+b\right)}\times 14+\left(-\frac{b}{2\left(9a+b\right)}\right)\left(-19\right)\\\frac{1}{9a+b}\times 14+\frac{a}{9a+b}\left(-19\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19b+126}{2\left(9a+b\right)}\\\frac{14-19a}{9a+b}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19b+126}{2\left(9a+b\right)},y=\frac{14-19a}{9a+b}$$

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.

$$2ax+by=14,-2x+9y=-19$$

Kia ōrite ai a 2ax me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2a.

$$-2\times 2ax-2by=-2\times 14,2a\left(-2\right)x+2a\times 9y=2a\left(-19\right)$$

Whakarūnātia.

$$\left(-4a\right)x+\left(-2b\right)y=-28,\left(-4a\right)x+18ay=-38a$$

Me tango \left(-4a\right)x+18ay=-38a mai i \left(-4a\right)x+\left(-2b\right)y=-28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\left(-4a\right)x+4ax+\left(-2b\right)y+\left(-18a\right)y=-28+38a$$

Tāpiri -4ax ki te 4ax. Ka whakakore atu ngā kupu -4ax me 4ax, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-2b\right)y+\left(-18a\right)y=-28+38a$$

Tāpiri -2by ki te -18ay.

$$\left(-18a-2b\right)y=-28+38a$$

Tāpiri -28 ki te 38a.

$$\left(-18a-2b\right)y=38a-28$$

Whakawehea ngā taha e rua ki te -2b-18a.

$$y=-\frac{19a-14}{9a+b}$$

Whakaurua te -\frac{-14+19a}{b+9a} mō y ki -2x+9y=-19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+9\left(-\frac{19a-14}{9a+b}\right)=-19$$

Whakareatia 9 ki te -\frac{-14+19a}{b+9a}.

$$-2x-\frac{9\left(19a-14\right)}{9a+b}=-19$$

Me tāpiri \frac{9\left(-14+19a\right)}{b+9a} ki ngā taha e rua o te whārite.

$$-2x=-\frac{19b+126}{9a+b}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{19b+126}{2\left(9a+b\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19b+126}{2\left(9a+b\right)},y=-\frac{19a-14}{9a+b}$$

Whakaoti mō x, y

$$\left\{\begin{matrix}x=\frac{19b+126}{2\left(9a+b\right)}\text{, }y=-\frac{19a-14}{9a+b}\text{, }&a\neq -\frac{b}{9}\\x=\frac{9y+19}{2}\text{, }y\in \mathrm{R}\text{, }&b=-\frac{126}{19}\text{ and }a=\frac{14}{19}\end{matrix}\right.$$

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.

$$2ax+by=14,-2x+9y=-19$$

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

$$2ax+by=14$$

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

$$2ax=\left(-b\right)y+14$$

Whakawehea ngā taha e rua ki te 2a.

$$x=\frac{1}{2a}\left(\left(-b\right)y+14\right)$$

Whakareatia \frac{1}{2a} ki te -by+14.

$$x=\left(-\frac{b}{2a}\right)y+\frac{7}{a}$$

Whakakapia te \frac{-by+14}{2a} mō te x ki tērā atu whārite, -2x+9y=-19.

$$-2\left(\left(-\frac{b}{2a}\right)y+\frac{7}{a}\right)+9y=-19$$

Whakareatia -2 ki te \frac{-by+14}{2a}.

$$\frac{b}{a}y-\frac{14}{a}+9y=-19$$

Tāpiri \frac{by}{a} ki te 9y.

$$\left(\frac{b}{a}+9\right)y-\frac{14}{a}=-19$$

Me tāpiri \frac{14}{a} ki ngā taha e rua o te whārite.

$$\left(\frac{b}{a}+9\right)y=-19+\frac{14}{a}$$

Whakawehea ngā taha e rua ki te 9+\frac{b}{a}.

$$y=\frac{14-19a}{9a+b}$$

Whakaurua te \frac{14-19a}{9a+b} mō y ki x=\left(-\frac{b}{2a}\right)y+\frac{7}{a}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{b}{2a}\right)\times \frac{14-19a}{9a+b}+\frac{7}{a}$$

Whakareatia -\frac{b}{2a} ki te \frac{14-19a}{9a+b}.

$$x=-\frac{b\left(14-19a\right)}{2a\left(9a+b\right)}+\frac{7}{a}$$

Tāpiri \frac{7}{a} ki te -\frac{b\left(14-19a\right)}{2a\left(9a+b\right)}.

$$x=\frac{19b+126}{2\left(9a+b\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19b+126}{2\left(9a+b\right)},y=\frac{14-19a}{9a+b}$$

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.

$$2ax+by=14,-2x+9y=-19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2a&b\\-2&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}14\\-19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}14\\-19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2a&b\\-2&9\end{matrix}\right))\left(\begin{matrix}14\\-19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2a\times 9-b\left(-2\right)}&-\frac{b}{2a\times 9-b\left(-2\right)}\\-\frac{-2}{2a\times 9-b\left(-2\right)}&\frac{2a}{2a\times 9-b\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}14\\-19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\left(9a+b\right)}&-\frac{b}{2\left(9a+b\right)}\\\frac{1}{9a+b}&\frac{a}{9a+b}\end{matrix}\right)\left(\begin{matrix}14\\-19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\left(9a+b\right)}\times 14+\left(-\frac{b}{2\left(9a+b\right)}\right)\left(-19\right)\\\frac{1}{9a+b}\times 14+\frac{a}{9a+b}\left(-19\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19b+126}{2\left(9a+b\right)}\\\frac{14-19a}{9a+b}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19b+126}{2\left(9a+b\right)},y=\frac{14-19a}{9a+b}$$

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.

$$2ax+by=14,-2x+9y=-19$$

Kia ōrite ai a 2ax me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2a.

$$-2\times 2ax-2by=-2\times 14,2a\left(-2\right)x+2a\times 9y=2a\left(-19\right)$$

Whakarūnātia.

$$\left(-4a\right)x+\left(-2b\right)y=-28,\left(-4a\right)x+18ay=-38a$$

Me tango \left(-4a\right)x+18ay=-38a mai i \left(-4a\right)x+\left(-2b\right)y=-28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\left(-4a\right)x+4ax+\left(-2b\right)y+\left(-18a\right)y=-28+38a$$

Tāpiri -4ax ki te 4ax. Ka whakakore atu ngā kupu -4ax me 4ax, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-2b\right)y+\left(-18a\right)y=-28+38a$$

Tāpiri -2by ki te -18ay.

$$\left(-18a-2b\right)y=-28+38a$$

Tāpiri -28 ki te 38a.

$$\left(-18a-2b\right)y=38a-28$$

Whakawehea ngā taha e rua ki te -2b-18a.

$$y=-\frac{19a-14}{9a+b}$$

Whakaurua te -\frac{-14+19a}{b+9a} mō y ki -2x+9y=-19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+9\left(-\frac{19a-14}{9a+b}\right)=-19$$

Whakareatia 9 ki te -\frac{-14+19a}{b+9a}.

$$-2x-\frac{9\left(19a-14\right)}{9a+b}=-19$$

Me tāpiri \frac{9\left(-14+19a\right)}{b+9a} ki ngā taha e rua o te whārite.

$$-2x=-\frac{19b+126}{9a+b}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{19b+126}{2\left(9a+b\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19b+126}{2\left(9a+b\right)},y=-\frac{19a-14}{9a+b}$$