Whakaoti i te {l}{-10x-3y=9}{-5x+5y=-2}

$$\left\{ \begin{array} { l } { - 10 x - 3 y = 9 } \\ { - 5 x + 5 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{3}{5}=-0.6$$

$$y=-1$$

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.

$$-10x-3y=9,-5x+5y=-2$$

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

$$-10x-3y=9$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$-10x=3y+9$$

Whakawehea ngā taha e rua ki te -10.

$$x=-\frac{1}{10}\left(3y+9\right)$$

Whakareatia -\frac{1}{10} ki te 9+3y.

$$x=-\frac{3}{10}y-\frac{9}{10}$$

Whakakapia te \frac{-3y-9}{10} mō te x ki tērā atu whārite, -5x+5y=-2.

$$-5\left(-\frac{3}{10}y-\frac{9}{10}\right)+5y=-2$$

Whakareatia -5 ki te \frac{-3y-9}{10}.

$$\frac{3}{2}y+\frac{9}{2}+5y=-2$$

Tāpiri \frac{3y}{2} ki te 5y.

$$\frac{13}{2}y+\frac{9}{2}=-2$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{2}, 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{3}{10}y-\frac{9}{10}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{10}\left(-1\right)-\frac{9}{10}$$

Whakareatia -\frac{3}{10} ki te -1.

$$x=\frac{3-9}{10}$$

Tāpiri -\frac{9}{10} ki te \frac{3}{10} 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.

$$x=-\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{5},y=-1$$

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.

$$-10x-3y=9,-5x+5y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right))\left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right))\left(\begin{matrix}9\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right))\left(\begin{matrix}9\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-3\\-5&5\end{matrix}\right))\left(\begin{matrix}9\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{-10\times 5-\left(-3\left(-5\right)\right)}&-\frac{-3}{-10\times 5-\left(-3\left(-5\right)\right)}\\-\frac{-5}{-10\times 5-\left(-3\left(-5\right)\right)}&-\frac{10}{-10\times 5-\left(-3\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}9\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&-\frac{3}{65}\\-\frac{1}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}9\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}\times 9-\frac{3}{65}\left(-2\right)\\-\frac{1}{13}\times 9+\frac{2}{13}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5}\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{3}{5},y=-1$$

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.

$$-10x-3y=9,-5x+5y=-2$$

Kia ōrite ai a -10x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -10.

$$-5\left(-10\right)x-5\left(-3\right)y=-5\times 9,-10\left(-5\right)x-10\times 5y=-10\left(-2\right)$$

Whakarūnātia.

$$50x+15y=-45,50x-50y=20$$

Me tango 50x-50y=20 mai i 50x+15y=-45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-50x+15y+50y=-45-20$$

Tāpiri 50x ki te -50x. Ka whakakore atu ngā kupu 50x me -50x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y+50y=-45-20$$

Tāpiri 15y ki te 50y.

$$65y=-45-20$$

Tāpiri -45 ki te -20.

$$65y=-65$$

Whakawehea ngā taha e rua ki te 65.

$$y=-1$$

Whakaurua te -1 mō y ki -5x+5y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+5\left(-1\right)=-2$$

Whakareatia 5 ki te -1.

$$-5x-5=-2$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$-5x=3$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{5},y=-1$$