Whakaoti i te {l}{-x-y=-2}{9x-2y=-15}

$$\left. \begin{array} { l } { - x - y = - 2 } \\ { 9 x - 2 y = - 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=3$$

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.

$$-x-y=-2,9x-2y=-15$$

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

$$-x-y=-2$$

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

$$-x=y-2$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(y-2\right)$$

Whakareatia -1 ki te y-2.

$$x=-y+2$$

Whakakapia te -y+2 mō te x ki tērā atu whārite, 9x-2y=-15.

$$9\left(-y+2\right)-2y=-15$$

Whakareatia 9 ki te -y+2.

$$-9y+18-2y=-15$$

Tāpiri -9y ki te -2y.

$$-11y+18=-15$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-11y=-33$$

Whakawehea ngā taha e rua ki te -11.

$$y=3$$

Whakaurua te 3 mō y ki x=-y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3+2$$

Tāpiri 2 ki te -3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$

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.

$$-x-y=-2,9x-2y=-15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\-15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right))\left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right))\left(\begin{matrix}-2\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right))\left(\begin{matrix}-2\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-1\\9&-2\end{matrix}\right))\left(\begin{matrix}-2\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-\left(-2\right)-\left(-9\right)}&-\frac{-1}{-\left(-2\right)-\left(-9\right)}\\-\frac{9}{-\left(-2\right)-\left(-9\right)}&-\frac{1}{-\left(-2\right)-\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}-2\\-15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{11}&\frac{1}{11}\\-\frac{9}{11}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}-2\\-15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{11}\left(-2\right)+\frac{1}{11}\left(-15\right)\\-\frac{9}{11}\left(-2\right)-\frac{1}{11}\left(-15\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=3$$

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.

$$-x-y=-2,9x-2y=-15$$

Kia ōrite ai a -x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$9\left(-1\right)x+9\left(-1\right)y=9\left(-2\right),-9x-\left(-2y\right)=-\left(-15\right)$$

Whakarūnātia.

$$-9x-9y=-18,-9x+2y=15$$

Me tango -9x+2y=15 mai i -9x-9y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x-9y-2y=-18-15$$

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

$$-9y-2y=-18-15$$

Tāpiri -9y ki te -2y.

$$-11y=-18-15$$

Tāpiri -18 ki te -15.

$$-11y=-33$$

Whakawehea ngā taha e rua ki te -11.

$$y=3$$

Whakaurua te 3 mō y ki 9x-2y=-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x-2\times 3=-15$$

Whakareatia -2 ki te 3.

$$9x-6=-15$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$9x=-9$$

Whakawehea ngā taha e rua ki te 9.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$