Whakaoti i te {l}{2x-19y=-10}{19x-18y=13}

$$\left. \begin{array} { l } { 2 x - 19 y = - 10 } \\ { 19 x - 18 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{427}{325} = 1\frac{102}{325} \approx 1.313846154$$

$$y=\frac{216}{325}\approx 0.664615385$$

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.

$$2x-19y=-10,19x-18y=13$$

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

$$2x-19y=-10$$

Me tāpiri 19y ki ngā taha e rua o te whārite.

$$2x=19y-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(19y-10\right)$$

Whakareatia \frac{1}{2} ki te 19y-10.

$$x=\frac{19}{2}y-5$$

Whakakapia te \frac{19y}{2}-5 mō te x ki tērā atu whārite, 19x-18y=13.

$$19\left(\frac{19}{2}y-5\right)-18y=13$$

Whakareatia 19 ki te \frac{19y}{2}-5.

$$\frac{361}{2}y-95-18y=13$$

Tāpiri \frac{361y}{2} ki te -18y.

$$\frac{325}{2}y-95=13$$

Me tāpiri 95 ki ngā taha e rua o te whārite.

$$\frac{325}{2}y=108$$

Whakawehea ngā taha e rua o te whārite ki te \frac{325}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{216}{325}$$

Whakaurua te \frac{216}{325} mō y ki x=\frac{19}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{19}{2}\times \frac{216}{325}-5$$

Whakareatia \frac{19}{2} ki te \frac{216}{325} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{2052}{325}-5$$

Tāpiri -5 ki te \frac{2052}{325}.

$$x=\frac{427}{325}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{427}{325},y=\frac{216}{325}$$

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.

$$2x-19y=-10,19x-18y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-19\\19&-18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-19\\19&-18\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-19\\19&-18\end{matrix}\right))\left(\begin{matrix}2&-19\\19&-18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-19\\19&-18\end{matrix}\right))\left(\begin{matrix}-10\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-19\\19&-18\end{matrix}\right))\left(\begin{matrix}-10\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-19\\19&-18\end{matrix}\right))\left(\begin{matrix}-10\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{2\left(-18\right)-\left(-19\times 19\right)}&-\frac{-19}{2\left(-18\right)-\left(-19\times 19\right)}\\-\frac{19}{2\left(-18\right)-\left(-19\times 19\right)}&\frac{2}{2\left(-18\right)-\left(-19\times 19\right)}\end{matrix}\right)\left(\begin{matrix}-10\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{325}&\frac{19}{325}\\-\frac{19}{325}&\frac{2}{325}\end{matrix}\right)\left(\begin{matrix}-10\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{325}\left(-10\right)+\frac{19}{325}\times 13\\-\frac{19}{325}\left(-10\right)+\frac{2}{325}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{427}{325}\\\frac{216}{325}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{427}{325},y=\frac{216}{325}$$

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.

$$2x-19y=-10,19x-18y=13$$

Kia ōrite ai a 2x me 19x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 19 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$19\times 2x+19\left(-19\right)y=19\left(-10\right),2\times 19x+2\left(-18\right)y=2\times 13$$

Whakarūnātia.

$$38x-361y=-190,38x-36y=26$$

Me tango 38x-36y=26 mai i 38x-361y=-190 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$38x-38x-361y+36y=-190-26$$

Tāpiri 38x ki te -38x. Ka whakakore atu ngā kupu 38x me -38x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-361y+36y=-190-26$$

Tāpiri -361y ki te 36y.

$$-325y=-190-26$$

Tāpiri -190 ki te -26.

$$-325y=-216$$

Whakawehea ngā taha e rua ki te -325.

$$y=\frac{216}{325}$$

Whakaurua te \frac{216}{325} mō y ki 19x-18y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$19x-18\times \frac{216}{325}=13$$

Whakareatia -18 ki te \frac{216}{325}.

$$19x-\frac{3888}{325}=13$$

Me tāpiri \frac{3888}{325} ki ngā taha e rua o te whārite.

$$19x=\frac{8113}{325}$$

Whakawehea ngā taha e rua ki te 19.

$$x=\frac{427}{325}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{427}{325},y=\frac{216}{325}$$