Whakaoti i te {l}{x+y=107}{4x+2y=284}hline0k

$$\left. \begin{array} { l } { x + y = 107 } \\ { 4 x + 2 y = 284 } \\ \hline 0 k \end{array} \right.$$

Whakaoti mō x, y

$$x=35$$

$$y=72$$

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.

$$x+y=107,4x+2y=284$$

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

$$x+y=107$$

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

$$x=-y+107$$

Whakakapia te -y+107 mō te x ki tērā atu whārite, 4x+2y=284.

$$4\left(-y+107\right)+2y=284$$

Whakareatia 4 ki te -y+107.

$$-4y+428+2y=284$$

Tāpiri -4y ki te 2y.

$$-2y+428=284$$

Me tango 428 mai i ngā taha e rua o te whārite.

$$-2y=-144$$

Whakawehea ngā taha e rua ki te -2.

$$y=72$$

Whakaurua te 72 mō y ki x=-y+107. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-72+107$$

Tāpiri 107 ki te -72.

$$x=35$$

Kua oti te pūnaha te whakatau.

$$x=35,y=72$$

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.

$$x+y=107,4x+2y=284$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}107\\284\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\4&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}107\\284\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}107\\284\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}107\\284\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-4}&-\frac{1}{2-4}\\-\frac{4}{2-4}&\frac{1}{2-4}\end{matrix}\right)\left(\begin{matrix}107\\284\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&\frac{1}{2}\\2&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}107\\284\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-107+\frac{1}{2}\times 284\\2\times 107-\frac{1}{2}\times 284\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}35\\72\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=35,y=72$$

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.

$$x+y=107,4x+2y=284$$

Kia ōrite ai a x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4x+4y=4\times 107,4x+2y=284$$

Whakarūnātia.

$$4x+4y=428,4x+2y=284$$

Me tango 4x+2y=284 mai i 4x+4y=428 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+4y-2y=428-284$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-2y=428-284$$

Tāpiri 4y ki te -2y.

$$2y=428-284$$

Tāpiri 428 ki te -284.

$$2y=144$$

Whakawehea ngā taha e rua ki te 2.

$$y=72$$

Whakaurua te 72 mō y ki 4x+2y=284. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+2\times 72=284$$

Whakareatia 2 ki te 72.

$$4x+144=284$$

Me tango 144 mai i ngā taha e rua o te whārite.

$$4x=140$$

Whakawehea ngā taha e rua ki te 4.

$$x=35$$

Kua oti te pūnaha te whakatau.

$$x=35,y=72$$