Whakaoti i te {c}{x+3y=14}{4x-y=4}

$$\left. \begin{array} { c } { x + 3 y = 14 } \\ { 4 x - y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=4$$

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.

$$x+3y=14,4x-y=4$$

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

$$x+3y=14$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$x=-3y+14$$

Whakakapia te -3y+14 mō te x ki tērā atu whārite, 4x-y=4.

$$4\left(-3y+14\right)-y=4$$

Whakareatia 4 ki te -3y+14.

$$-12y+56-y=4$$

Tāpiri -12y ki te -y.

$$-13y+56=4$$

Me tango 56 mai i ngā taha e rua o te whārite.

$$-13y=-52$$

Whakawehea ngā taha e rua ki te -13.

$$y=4$$

Whakaurua te 4 mō y ki x=-3y+14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\times 4+14$$

Whakareatia -3 ki te 4.

$$x=-12+14$$

Tāpiri 14 ki te -12.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$

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.

$$x+3y=14,4x-y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&3\\4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&3\\4&-1\end{matrix}\right))\left(\begin{matrix}1&3\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\4&-1\end{matrix}\right))\left(\begin{matrix}14\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\4&-1\end{matrix}\right))\left(\begin{matrix}14\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\4&-1\end{matrix}\right))\left(\begin{matrix}14\\4\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-3\times 4}&-\frac{3}{-1-3\times 4}\\-\frac{4}{-1-3\times 4}&\frac{1}{-1-3\times 4}\end{matrix}\right)\left(\begin{matrix}14\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{3}{13}\\\frac{4}{13}&-\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}14\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\times 14+\frac{3}{13}\times 4\\\frac{4}{13}\times 14-\frac{1}{13}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=4$$

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.

$$x+3y=14,4x-y=4$$

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.

$$4x+4\times 3y=4\times 14,4x-y=4$$

Whakarūnātia.

$$4x+12y=56,4x-y=4$$

Me tango 4x-y=4 mai i 4x+12y=56 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+12y+y=56-4$$

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.

$$12y+y=56-4$$

Tāpiri 12y ki te y.

$$13y=56-4$$

Tāpiri 56 ki te -4.

$$13y=52$$

Whakawehea ngā taha e rua ki te 13.

$$y=4$$

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

$$4x-4=4$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$