Whakaoti i te {l}{11a+55d=132}{2a+13d=30}

$$\left\{ \begin{array} { l } { 11 a + 55 d = 132 } \\ { 2 a + 13 d = 30 } \end{array} \right.$$

Whakaoti mō a, d

$$a=2$$

$$d=2$$

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.

$$11a+55d=132,2a+13d=30$$

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

$$11a+55d=132$$

Me tango 55d mai i ngā taha e rua o te whārite.

$$11a=-55d+132$$

Whakawehea ngā taha e rua ki te 11.

$$a=\frac{1}{11}\left(-55d+132\right)$$

Whakareatia \frac{1}{11} ki te -55d+132.

$$a=-5d+12$$

Whakakapia te -5d+12 mō te a ki tērā atu whārite, 2a+13d=30.

$$2\left(-5d+12\right)+13d=30$$

Whakareatia 2 ki te -5d+12.

$$-10d+24+13d=30$$

Tāpiri -10d ki te 13d.

$$3d+24=30$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$3d=6$$

Whakawehea ngā taha e rua ki te 3.

$$d=2$$

Whakaurua te 2 mō d ki a=-5d+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-5\times 2+12$$

Whakareatia -5 ki te 2.

$$a=-10+12$$

Tāpiri 12 ki te -10.

$$a=2$$

Kua oti te pūnaha te whakatau.

$$a=2,d=2$$

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.

$$11a+55d=132,2a+13d=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}11&55\\2&13\end{matrix}\right)\left(\begin{matrix}a\\d\end{matrix}\right)=\left(\begin{matrix}132\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}11&55\\2&13\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&55\\2&13\end{matrix}\right))\left(\begin{matrix}11&55\\2&13\end{matrix}\right)\left(\begin{matrix}a\\d\end{matrix}\right)=inverse(\left(\begin{matrix}11&55\\2&13\end{matrix}\right))\left(\begin{matrix}132\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\d\end{matrix}\right)=inverse(\left(\begin{matrix}11&55\\2&13\end{matrix}\right))\left(\begin{matrix}132\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\d\end{matrix}\right)=inverse(\left(\begin{matrix}11&55\\2&13\end{matrix}\right))\left(\begin{matrix}132\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\d\end{matrix}\right)=\left(\begin{matrix}\frac{13}{11\times 13-55\times 2}&-\frac{55}{11\times 13-55\times 2}\\-\frac{2}{11\times 13-55\times 2}&\frac{11}{11\times 13-55\times 2}\end{matrix}\right)\left(\begin{matrix}132\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\d\end{matrix}\right)=\left(\begin{matrix}\frac{13}{33}&-\frac{5}{3}\\-\frac{2}{33}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}132\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\d\end{matrix}\right)=\left(\begin{matrix}\frac{13}{33}\times 132-\frac{5}{3}\times 30\\-\frac{2}{33}\times 132+\frac{1}{3}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\d\end{matrix}\right)=\left(\begin{matrix}2\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me d.

$$a=2,d=2$$

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.

$$11a+55d=132,2a+13d=30$$

Kia ōrite ai a 11a me 2a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 11.

$$2\times 11a+2\times 55d=2\times 132,11\times 2a+11\times 13d=11\times 30$$

Whakarūnātia.

$$22a+110d=264,22a+143d=330$$

Me tango 22a+143d=330 mai i 22a+110d=264 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$22a-22a+110d-143d=264-330$$

Tāpiri 22a ki te -22a. Ka whakakore atu ngā kupu 22a me -22a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$110d-143d=264-330$$

Tāpiri 110d ki te -143d.

$$-33d=264-330$$

Tāpiri 264 ki te -330.

$$-33d=-66$$

Whakawehea ngā taha e rua ki te -33.

$$d=2$$

Whakaurua te 2 mō d ki 2a+13d=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$2a+13\times 2=30$$

Whakareatia 13 ki te 2.

$$2a+26=30$$

Me tango 26 mai i ngā taha e rua o te whārite.

$$2a=4$$

Whakawehea ngā taha e rua ki te 2.

$$a=2$$

Kua oti te pūnaha te whakatau.

$$a=2,d=2$$