Whakaoti i te {c}{x+y=64}{12x+26y=19}

$$\left. \begin{array} { c } { x + y = 64 } \\ { 12 x + 26 y = 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{235}{2} = 117\frac{1}{2} = 117.5$$

$$y = -\frac{107}{2} = -53\frac{1}{2} = -53.5$$

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.

$$x+y=64,12x+26y=19$$

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

$$x+y=64$$

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

$$x=-y+64$$

Whakakapia te -y+64 mō te x ki tērā atu whārite, 12x+26y=19.

$$12\left(-y+64\right)+26y=19$$

Whakareatia 12 ki te -y+64.

$$-12y+768+26y=19$$

Tāpiri -12y ki te 26y.

$$14y+768=19$$

Me tango 768 mai i ngā taha e rua o te whārite.

$$14y=-749$$

Whakawehea ngā taha e rua ki te 14.

$$y=-\frac{107}{2}$$

Whakaurua te -\frac{107}{2} mō y ki x=-y+64. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-\frac{107}{2}\right)+64$$

Whakareatia -1 ki te -\frac{107}{2}.

$$x=\frac{107}{2}+64$$

Tāpiri 64 ki te \frac{107}{2}.

$$x=\frac{235}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{235}{2},y=-\frac{107}{2}$$

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.

$$x+y=64,12x+26y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\12&26\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}64\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\12&26\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\12&26\end{matrix}\right))\left(\begin{matrix}1&1\\12&26\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\12&26\end{matrix}\right))\left(\begin{matrix}64\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\12&26\end{matrix}\right))\left(\begin{matrix}64\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\12&26\end{matrix}\right))\left(\begin{matrix}64\\19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{26}{26-12}&-\frac{1}{26-12}\\-\frac{12}{26-12}&\frac{1}{26-12}\end{matrix}\right)\left(\begin{matrix}64\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{7}&-\frac{1}{14}\\-\frac{6}{7}&\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}64\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{7}\times 64-\frac{1}{14}\times 19\\-\frac{6}{7}\times 64+\frac{1}{14}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{235}{2}\\-\frac{107}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{235}{2},y=-\frac{107}{2}$$

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.

$$x+y=64,12x+26y=19$$

Kia ōrite ai a x me 12x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$12x+12y=12\times 64,12x+26y=19$$

Whakarūnātia.

$$12x+12y=768,12x+26y=19$$

Me tango 12x+26y=19 mai i 12x+12y=768 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+12y-26y=768-19$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-26y=768-19$$

Tāpiri 12y ki te -26y.

$$-14y=768-19$$

Tāpiri 768 ki te -19.

$$-14y=749$$

Whakawehea ngā taha e rua ki te -14.

$$y=-\frac{107}{2}$$

Whakaurua te -\frac{107}{2} mō y ki 12x+26y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$12x+26\left(-\frac{107}{2}\right)=19$$

Whakareatia 26 ki te -\frac{107}{2}.

$$12x-1391=19$$

Me tāpiri 1391 ki ngā taha e rua o te whārite.

$$12x=1410$$

Whakawehea ngā taha e rua ki te 12.

$$x=\frac{235}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{235}{2},y=-\frac{107}{2}$$