Whakaoti i te {l}{3x-7y=-19}{2x-9y=-17}

$$\left. \begin{array} { l } { 3 x - 7 y = - 19 } \\ { 2 x - 9 y = - 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=1$$

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.

$$3x-7y=-19,2x-9y=-17$$

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

$$3x-7y=-19$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$3x=7y-19$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(7y-19\right)$$

Whakareatia \frac{1}{3} ki te 7y-19.

$$x=\frac{7}{3}y-\frac{19}{3}$$

Whakakapia te \frac{7y-19}{3} mō te x ki tērā atu whārite, 2x-9y=-17.

$$2\left(\frac{7}{3}y-\frac{19}{3}\right)-9y=-17$$

Whakareatia 2 ki te \frac{7y-19}{3}.

$$\frac{14}{3}y-\frac{38}{3}-9y=-17$$

Tāpiri \frac{14y}{3} ki te -9y.

$$-\frac{13}{3}y-\frac{38}{3}=-17$$

Me tāpiri \frac{38}{3} ki ngā taha e rua o te whārite.

$$-\frac{13}{3}y=-\frac{13}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{7}{3}y-\frac{19}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7-19}{3}$$

Tāpiri -\frac{19}{3} ki te \frac{7}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=1$$

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.

$$3x-7y=-19,2x-9y=-17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-7\\2&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-19\\-17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-7\\2&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-7\\2&-9\end{matrix}\right))\left(\begin{matrix}3&-7\\2&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\2&-9\end{matrix}\right))\left(\begin{matrix}-19\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\2&-9\end{matrix}\right))\left(\begin{matrix}-19\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\2&-9\end{matrix}\right))\left(\begin{matrix}-19\\-17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{3\left(-9\right)-\left(-7\times 2\right)}&-\frac{-7}{3\left(-9\right)-\left(-7\times 2\right)}\\-\frac{2}{3\left(-9\right)-\left(-7\times 2\right)}&\frac{3}{3\left(-9\right)-\left(-7\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-19\\-17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{13}&-\frac{7}{13}\\\frac{2}{13}&-\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}-19\\-17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{13}\left(-19\right)-\frac{7}{13}\left(-17\right)\\\frac{2}{13}\left(-19\right)-\frac{3}{13}\left(-17\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=1$$

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.

$$3x-7y=-19,2x-9y=-17$$

Kia ōrite ai a 3x 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 3.

$$2\times 3x+2\left(-7\right)y=2\left(-19\right),3\times 2x+3\left(-9\right)y=3\left(-17\right)$$

Whakarūnātia.

$$6x-14y=-38,6x-27y=-51$$

Me tango 6x-27y=-51 mai i 6x-14y=-38 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-14y+27y=-38+51$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-14y+27y=-38+51$$

Tāpiri -14y ki te 27y.

$$13y=-38+51$$

Tāpiri -38 ki te 51.

$$13y=13$$

Whakawehea ngā taha e rua ki te 13.

$$y=1$$

Whakaurua te 1 mō y ki 2x-9y=-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-9=-17$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$2x=-8$$

Whakawehea ngā taha e rua ki te 2.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=1$$