Whakaoti i te {l}{x+3y=26}{7x-2y=44}

$$\left. \begin{array} { l } { x + 3 y = 26 } \\ { 7 x - 2 y = 44 } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=6$$

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.

$$x+3y=26,7x-2y=44$$

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

$$x+3y=26$$

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

$$x=-3y+26$$

Whakakapia te -3y+26 mō te x ki tērā atu whārite, 7x-2y=44.

$$7\left(-3y+26\right)-2y=44$$

Whakareatia 7 ki te -3y+26.

$$-21y+182-2y=44$$

Tāpiri -21y ki te -2y.

$$-23y+182=44$$

Me tango 182 mai i ngā taha e rua o te whārite.

$$-23y=-138$$

Whakawehea ngā taha e rua ki te -23.

$$y=6$$

Whakaurua te 6 mō y ki x=-3y+26. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\times 6+26$$

Whakareatia -3 ki te 6.

$$x=-18+26$$

Tāpiri 26 ki te -18.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=6$$

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.

$$x+3y=26,7x-2y=44$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\7&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}26\\44\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&3\\7&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&3\\7&-2\end{matrix}\right))\left(\begin{matrix}1&3\\7&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\7&-2\end{matrix}\right))\left(\begin{matrix}26\\44\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\7&-2\end{matrix}\right))\left(\begin{matrix}26\\44\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\7&-2\end{matrix}\right))\left(\begin{matrix}26\\44\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-3\times 7}&-\frac{3}{-2-3\times 7}\\-\frac{7}{-2-3\times 7}&\frac{1}{-2-3\times 7}\end{matrix}\right)\left(\begin{matrix}26\\44\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{23}&\frac{3}{23}\\\frac{7}{23}&-\frac{1}{23}\end{matrix}\right)\left(\begin{matrix}26\\44\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{23}\times 26+\frac{3}{23}\times 44\\\frac{7}{23}\times 26-\frac{1}{23}\times 44\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=6$$

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.

$$x+3y=26,7x-2y=44$$

Kia ōrite ai a x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$7x+7\times 3y=7\times 26,7x-2y=44$$

Whakarūnātia.

$$7x+21y=182,7x-2y=44$$

Me tango 7x-2y=44 mai i 7x+21y=182 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x+21y+2y=182-44$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y+2y=182-44$$

Tāpiri 21y ki te 2y.

$$23y=182-44$$

Tāpiri 182 ki te -44.

$$23y=138$$

Whakawehea ngā taha e rua ki te 23.

$$y=6$$

Whakaurua te 6 mō y ki 7x-2y=44. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-2\times 6=44$$

Whakareatia -2 ki te 6.

$$7x-12=44$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$7x=56$$

Whakawehea ngā taha e rua ki te 7.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=6$$