Whakaoti i te {l}{-3x+3y=-3}{x-9y=-15}

$$\left. \begin{array} { l } { - 3 x + 3 y = - 3 } \\ { x - 9 y = - 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=2$$

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.

$$-3x+3y=-3,x-9y=-15$$

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

$$-3x+3y=-3$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$-3x=-3y-3$$

Whakawehea ngā taha e rua ki te -3.

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

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

$$x=y+1$$

Whakakapia te y+1 mō te x ki tērā atu whārite, x-9y=-15.

$$y+1-9y=-15$$

Tāpiri y ki te -9y.

$$-8y+1=-15$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-8y=-16$$

Whakawehea ngā taha e rua ki te -8.

$$y=2$$

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

$$x=2+1$$

Tāpiri 1 ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$

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.

$$-3x+3y=-3,x-9y=-15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&3\\1&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&3\\1&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&3\\1&-9\end{matrix}\right))\left(\begin{matrix}-3&3\\1&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&3\\1&-9\end{matrix}\right))\left(\begin{matrix}-3\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&3\\1&-9\end{matrix}\right))\left(\begin{matrix}-3\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&3\\1&-9\end{matrix}\right))\left(\begin{matrix}-3\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{-3\left(-9\right)-3}&-\frac{3}{-3\left(-9\right)-3}\\-\frac{1}{-3\left(-9\right)-3}&-\frac{3}{-3\left(-9\right)-3}\end{matrix}\right)\left(\begin{matrix}-3\\-15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{8}&-\frac{1}{8}\\-\frac{1}{24}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}-3\\-15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{8}\left(-3\right)-\frac{1}{8}\left(-15\right)\\-\frac{1}{24}\left(-3\right)-\frac{1}{8}\left(-15\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=2$$

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.

$$-3x+3y=-3,x-9y=-15$$

Kia ōrite ai a -3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

$$-3x+3y=-3,-3x-3\left(-9\right)y=-3\left(-15\right)$$

Whakarūnātia.

$$-3x+3y=-3,-3x+27y=45$$

Me tango -3x+27y=45 mai i -3x+3y=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+3y-27y=-3-45$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-27y=-3-45$$

Tāpiri 3y ki te -27y.

$$-24y=-3-45$$

Tāpiri -3 ki te -45.

$$-24y=-48$$

Whakawehea ngā taha e rua ki te -24.

$$y=2$$

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

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

Whakareatia -9 ki te 2.

$$x-18=-15$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$