Whakaoti i te {l}{-6x+6y=-6}{2x-y=-4}

$$\left. \begin{array} { l } { - 6 x + 6 y = - 6 } \\ { 2 x - y = - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=-6$$

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.

$$-6x+6y=-6,2x-y=-4$$

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

$$-6x+6y=-6$$

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

$$-6x=-6y-6$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{1}{6}\left(-6y-6\right)$$

Whakareatia -\frac{1}{6} ki te -6y-6.

$$x=y+1$$

Whakakapia te y+1 mō te x ki tērā atu whārite, 2x-y=-4.

$$2\left(y+1\right)-y=-4$$

Whakareatia 2 ki te y+1.

$$2y+2-y=-4$$

Tāpiri 2y ki te -y.

$$y+2=-4$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$y=-6$$

Whakaurua te -6 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.

$$x=-6+1$$

Tāpiri 1 ki te -6.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-6$$

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.

$$-6x+6y=-6,2x-y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&6\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-6&6\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&6\\2&-1\end{matrix}\right))\left(\begin{matrix}-6&6\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&6\\2&-1\end{matrix}\right))\left(\begin{matrix}-6\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&6\\2&-1\end{matrix}\right))\left(\begin{matrix}-6\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&6\\2&-1\end{matrix}\right))\left(\begin{matrix}-6\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-6\left(-1\right)-6\times 2}&-\frac{6}{-6\left(-1\right)-6\times 2}\\-\frac{2}{-6\left(-1\right)-6\times 2}&-\frac{6}{-6\left(-1\right)-6\times 2}\end{matrix}\right)\left(\begin{matrix}-6\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&1\\\frac{1}{3}&1\end{matrix}\right)\left(\begin{matrix}-6\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\left(-6\right)-4\\\frac{1}{3}\left(-6\right)-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=-6$$

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.

$$-6x+6y=-6,2x-y=-4$$

Kia ōrite ai a -6x 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 -6.

$$2\left(-6\right)x+2\times 6y=2\left(-6\right),-6\times 2x-6\left(-1\right)y=-6\left(-4\right)$$

Whakarūnātia.

$$-12x+12y=-12,-12x+6y=24$$

Me tango -12x+6y=24 mai i -12x+12y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x+12y-6y=-12-24$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-6y=-12-24$$

Tāpiri 12y ki te -6y.

$$6y=-12-24$$

Tāpiri -12 ki te -24.

$$6y=-36$$

Whakawehea ngā taha e rua ki te 6.

$$y=-6$$

Whakaurua te -6 mō y ki 2x-y=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-\left(-6\right)=-4$$

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

$$2x=-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-6$$