Whakaoti i te {l}{x+y=50}{x+2y=87}

$$\left. \begin{array} { l } { x + y = 50 } \\ { x + 2 y = 87 } \end{array} \right.$$

Whakaoti mō x, y

$$x=13$$

$$y=37$$

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.

$$x+y=50,x+2y=87$$

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

$$x+y=50$$

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

$$x=-y+50$$

Whakakapia te -y+50 mō te x ki tērā atu whārite, x+2y=87.

$$-y+50+2y=87$$

Tāpiri -y ki te 2y.

$$y+50=87$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$y=37$$

Whakaurua te 37 mō y ki x=-y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-37+50$$

Tāpiri 50 ki te -37.

$$x=13$$

Kua oti te pūnaha te whakatau.

$$x=13,y=37$$

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.

$$x+y=50,x+2y=87$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\87\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}1&1\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}50\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}50\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&2\end{matrix}\right))\left(\begin{matrix}50\\87\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-1}&-\frac{1}{2-1}\\-\frac{1}{2-1}&\frac{1}{2-1}\end{matrix}\right)\left(\begin{matrix}50\\87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right)\left(\begin{matrix}50\\87\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 50-87\\-50+87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\37\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=13,y=37$$

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.

$$x+y=50,x+2y=87$$

Me tango x+2y=87 mai i x+y=50 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y-2y=50-87$$

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.

$$y-2y=50-87$$

Tāpiri y ki te -2y.

$$-y=50-87$$

Tāpiri 50 ki te -87.

$$-y=-37$$

Whakawehea ngā taha e rua ki te -1.

$$y=37$$

Whakaurua te 37 mō y ki x+2y=87. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2\times 37=87$$

Whakareatia 2 ki te 37.

$$x+74=87$$

Me tango 74 mai i ngā taha e rua o te whārite.

$$x=13$$

Kua oti te pūnaha te whakatau.

$$x=13,y=37$$