Whakaoti i te {l}{-3x+3y=-9}{6x-y=-12}

$$\left. \begin{array} { l } { - 3 x + 3 y = - 9 } \\ { 6 x - y = - 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-6$$

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.

$$-3x+3y=-9,6x-y=-12$$

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

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

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

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

Whakawehea ngā taha e rua ki te -3.

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

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

$$x=y+3$$

Whakakapia te y+3 mō te x ki tērā atu whārite, 6x-y=-12.

$$6\left(y+3\right)-y=-12$$

Whakareatia 6 ki te y+3.

$$6y+18-y=-12$$

Tāpiri 6y ki te -y.

$$5y+18=-12$$

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

$$5y=-30$$

Whakawehea ngā taha e rua ki te 5.

$$y=-6$$

Whakaurua te -6 mō y ki x=y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6+3$$

Tāpiri 3 ki te -6.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-6$$

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.

$$-3x+3y=-9,6x-y=-12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&3\\6&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&3\\6&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&3\\6&-1\end{matrix}\right))\left(\begin{matrix}-3&3\\6&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&3\\6&-1\end{matrix}\right))\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&3\\6&-1\end{matrix}\right))\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&3\\6&-1\end{matrix}\right))\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-3\left(-1\right)-3\times 6}&-\frac{3}{-3\left(-1\right)-3\times 6}\\-\frac{6}{-3\left(-1\right)-3\times 6}&-\frac{3}{-3\left(-1\right)-3\times 6}\end{matrix}\right)\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&\frac{1}{5}\\\frac{2}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\left(-9\right)+\frac{1}{5}\left(-12\right)\\\frac{2}{5}\left(-9\right)+\frac{1}{5}\left(-12\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-6$$

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.

$$-3x+3y=-9,6x-y=-12$$

Kia ōrite ai a -3x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

$$6\left(-3\right)x+6\times 3y=6\left(-9\right),-3\times 6x-3\left(-1\right)y=-3\left(-12\right)$$

Whakarūnātia.

$$-18x+18y=-54,-18x+3y=36$$

Me tango -18x+3y=36 mai i -18x+18y=-54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-18x+18x+18y-3y=-54-36$$

Tāpiri -18x ki te 18x. Ka whakakore atu ngā kupu -18x me 18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y-3y=-54-36$$

Tāpiri 18y ki te -3y.

$$15y=-54-36$$

Tāpiri -54 ki te -36.

$$15y=-90$$

Whakawehea ngā taha e rua ki te 15.

$$y=-6$$

Whakaurua te -6 mō y ki 6x-y=-12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-\left(-6\right)=-12$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$6x=-18$$

Whakawehea ngā taha e rua ki te 6.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-6$$