Whakaoti i te {l}{x-6y=3}{2x-18y=-6}

$$\left\{ \begin{array} { l } { x - 6 y = 3 } \\ { 2 x - 18 y = - 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=15$$

$$y=2$$

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.

$$x-6y=3,2x-18y=-6$$

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

$$x-6y=3$$

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

$$x=6y+3$$

Whakakapia te 6y+3 mō te x ki tērā atu whārite, 2x-18y=-6.

$$2\left(6y+3\right)-18y=-6$$

Whakareatia 2 ki te 6y+3.

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

Tāpiri 12y ki te -18y.

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

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

$$-6y=-12$$

Whakawehea ngā taha e rua ki te -6.

$$y=2$$

Whakaurua te 2 mō y ki x=6y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=6\times 2+3$$

Whakareatia 6 ki te 2.

$$x=12+3$$

Tāpiri 3 ki te 12.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=2$$

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.

$$x-6y=3,2x-18y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-6\\2&-18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-6\\2&-18\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-6\\2&-18\end{matrix}\right))\left(\begin{matrix}1&-6\\2&-18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\2&-18\end{matrix}\right))\left(\begin{matrix}3\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\2&-18\end{matrix}\right))\left(\begin{matrix}3\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-6\\2&-18\end{matrix}\right))\left(\begin{matrix}3\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{-18-\left(-6\times 2\right)}&-\frac{-6}{-18-\left(-6\times 2\right)}\\-\frac{2}{-18-\left(-6\times 2\right)}&\frac{1}{-18-\left(-6\times 2\right)}\end{matrix}\right)\left(\begin{matrix}3\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=15,y=2$$

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.

$$x-6y=3,2x-18y=-6$$

Kia ōrite ai a x 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 1.

$$2x+2\left(-6\right)y=2\times 3,2x-18y=-6$$

Whakarūnātia.

$$2x-12y=6,2x-18y=-6$$

Me tango 2x-18y=-6 mai i 2x-12y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-12y+18y=6+6$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -12y ki te 18y.

$$6y=6+6$$

Tāpiri 6 ki te 6.

$$6y=12$$

Whakawehea ngā taha e rua ki te 6.

$$y=2$$

Whakaurua te 2 mō y ki 2x-18y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-18\times 2=-6$$

Whakareatia -18 ki te 2.

$$2x-36=-6$$

Me tāpiri 36 ki ngā taha e rua o te whārite.

$$2x=30$$

Whakawehea ngā taha e rua ki te 2.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=2$$