Whakaoti i te {l}{x+6y=27}{7x-3y=9}

$$\left. \begin{array} { l } { x + 6 y = 27 } \\ { 7 x - 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=4$$

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.

$$x+6y=27,7x-3y=9$$

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

$$x+6y=27$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$x=-6y+27$$

Whakakapia te -6y+27 mō te x ki tērā atu whārite, 7x-3y=9.

$$7\left(-6y+27\right)-3y=9$$

Whakareatia 7 ki te -6y+27.

$$-42y+189-3y=9$$

Tāpiri -42y ki te -3y.

$$-45y+189=9$$

Me tango 189 mai i ngā taha e rua o te whārite.

$$-45y=-180$$

Whakawehea ngā taha e rua ki te -45.

$$y=4$$

Whakaurua te 4 mō y ki x=-6y+27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6\times 4+27$$

Whakareatia -6 ki te 4.

$$x=-24+27$$

Tāpiri 27 ki te -24.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$

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.

$$x+6y=27,7x-3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&6\\7&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&6\\7&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&6\\7&-3\end{matrix}\right))\left(\begin{matrix}1&6\\7&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\7&-3\end{matrix}\right))\left(\begin{matrix}27\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\7&-3\end{matrix}\right))\left(\begin{matrix}27\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\7&-3\end{matrix}\right))\left(\begin{matrix}27\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-6\times 7}&-\frac{6}{-3-6\times 7}\\-\frac{7}{-3-6\times 7}&\frac{1}{-3-6\times 7}\end{matrix}\right)\left(\begin{matrix}27\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&\frac{2}{15}\\\frac{7}{45}&-\frac{1}{45}\end{matrix}\right)\left(\begin{matrix}27\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 27+\frac{2}{15}\times 9\\\frac{7}{45}\times 27-\frac{1}{45}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=4$$

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.

$$x+6y=27,7x-3y=9$$

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.

$$7x+7\times 6y=7\times 27,7x-3y=9$$

Whakarūnātia.

$$7x+42y=189,7x-3y=9$$

Me tango 7x-3y=9 mai i 7x+42y=189 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x+42y+3y=189-9$$

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.

$$42y+3y=189-9$$

Tāpiri 42y ki te 3y.

$$45y=189-9$$

Tāpiri 189 ki te -9.

$$45y=180$$

Whakawehea ngā taha e rua ki te 45.

$$y=4$$

Whakaurua te 4 mō y ki 7x-3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-3\times 4=9$$

Whakareatia -3 ki te 4.

$$7x-12=9$$

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

$$7x=21$$

Whakawehea ngā taha e rua ki te 7.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$