Whakaoti i te {c}{x+y=11}{2y+x=17}

$$\left. \begin{array} { c } { x + y = 11 } \\ { 2 y + x = 17 } \end{array} \right.$$

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.

$$x+y=11,x+2y=17$$

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

$$x+y=11$$

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

$$x=-y+11$$

Whakakapia te -y+11 mō te x ki tērā atu whārite, x+2y=17.

$$-y+11+2y=17$$

Tāpiri -y ki te 2y.

$$y+11=17$$

Me tango 11 mai i ngā taha e rua o te whārite.

$$y=6$$

Whakaurua te 6 mō y ki x=-y+11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6+11$$

Tāpiri 11 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.

$$x+y=11,x+2y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\17\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}1&1\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}11\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}11\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}11\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-1}&-\frac{1}{2-1}\\-\frac{1}{2-1}&\frac{1}{2-1}\end{matrix}\right)\left(\begin{matrix}11\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right)\left(\begin{matrix}11\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 11-17\\-11+17\end{matrix}\right)$$

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.

$$x+y=11,x+2y=17$$

Me tango x+2y=17 mai i x+y=11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y-2y=11-17$$

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

$$y-2y=11-17$$

Tāpiri y ki te -2y.

$$-y=11-17$$

Tāpiri 11 ki te -17.

$$-y=-6$$

Whakawehea ngā taha e rua ki te -1.

$$y=6$$

Whakaurua te 6 mō y ki x+2y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2\times 6=17$$

Whakareatia 2 ki te 6.

$$x+12=17$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=6$$