Whakaoti i te {l}{4a_{1}+6d=3}{3a_{1}+21d=4}

$$\left. \begin{array} { l } { 4 a _ { 1 } + 6 d = 3 } \\ { 3 a _ { 1 } + 21 d = 4 } \end{array} \right.$$

Whakaoti mō a_1, d

$$a_{1}=\frac{13}{22}\approx 0.590909091$$

$$d=\frac{7}{66}\approx 0.106060606$$

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.

$$4a_{1}+6d=3,3a_{1}+21d=4$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te a_{1} mā te wehe i te a_{1} i te taha mauī o te tohu ōrite.

$$4a_{1}+6d=3$$

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

$$4a_{1}=-6d+3$$

Whakawehea ngā taha e rua ki te 4.

$$a_{1}=\frac{1}{4}\left(-6d+3\right)$$

Whakareatia \frac{1}{4} ki te -6d+3.

$$a_{1}=-\frac{3}{2}d+\frac{3}{4}$$

Whakakapia te -\frac{3d}{2}+\frac{3}{4} mō te a_{1} ki tērā atu whārite, 3a_{1}+21d=4.

$$3\left(-\frac{3}{2}d+\frac{3}{4}\right)+21d=4$$

Whakareatia 3 ki te -\frac{3d}{2}+\frac{3}{4}.

$$-\frac{9}{2}d+\frac{9}{4}+21d=4$$

Tāpiri -\frac{9d}{2} ki te 21d.

$$\frac{33}{2}d+\frac{9}{4}=4$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$\frac{33}{2}d=\frac{7}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{33}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$d=\frac{7}{66}$$

Whakaurua te \frac{7}{66} mō d ki a_{1}=-\frac{3}{2}d+\frac{3}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a_{1} hāngai tonu.

$$a_{1}=-\frac{3}{2}\times \frac{7}{66}+\frac{3}{4}$$

Whakareatia -\frac{3}{2} ki te \frac{7}{66} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a_{1}=-\frac{7}{44}+\frac{3}{4}$$

Tāpiri \frac{3}{4} ki te -\frac{7}{44} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a_{1}=\frac{13}{22}$$

Kua oti te pūnaha te whakatau.

$$a_{1}=\frac{13}{22},d=\frac{7}{66}$$

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.

$$4a_{1}+6d=3,3a_{1}+21d=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&6\\3&21\end{matrix}\right)\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}3\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&6\\3&21\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&6\\3&21\end{matrix}\right))\left(\begin{matrix}4&6\\3&21\end{matrix}\right)\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=inverse(\left(\begin{matrix}4&6\\3&21\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=inverse(\left(\begin{matrix}4&6\\3&21\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=inverse(\left(\begin{matrix}4&6\\3&21\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{21}{4\times 21-6\times 3}&-\frac{6}{4\times 21-6\times 3}\\-\frac{3}{4\times 21-6\times 3}&\frac{4}{4\times 21-6\times 3}\end{matrix}\right)\left(\begin{matrix}3\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{7}{22}&-\frac{1}{11}\\-\frac{1}{22}&\frac{2}{33}\end{matrix}\right)\left(\begin{matrix}3\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{7}{22}\times 3-\frac{1}{11}\times 4\\-\frac{1}{22}\times 3+\frac{2}{33}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{13}{22}\\\frac{7}{66}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a_{1} me d.

$$a_{1}=\frac{13}{22},d=\frac{7}{66}$$

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.

$$4a_{1}+6d=3,3a_{1}+21d=4$$

Kia ōrite ai a 4a_{1} me 3a_{1}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$3\times 4a_{1}+3\times 6d=3\times 3,4\times 3a_{1}+4\times 21d=4\times 4$$

Whakarūnātia.

$$12a_{1}+18d=9,12a_{1}+84d=16$$

Me tango 12a_{1}+84d=16 mai i 12a_{1}+18d=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12a_{1}-12a_{1}+18d-84d=9-16$$

Tāpiri 12a_{1} ki te -12a_{1}. Ka whakakore atu ngā kupu 12a_{1} me -12a_{1}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18d-84d=9-16$$

Tāpiri 18d ki te -84d.

$$-66d=9-16$$

Tāpiri 9 ki te -16.

$$-66d=-7$$

Whakawehea ngā taha e rua ki te -66.

$$d=\frac{7}{66}$$

Whakaurua te \frac{7}{66} mō d ki 3a_{1}+21d=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a_{1} hāngai tonu.

$$3a_{1}+21\times \frac{7}{66}=4$$

Whakareatia 21 ki te \frac{7}{66}.

$$3a_{1}+\frac{49}{22}=4$$

Me tango \frac{49}{22} mai i ngā taha e rua o te whārite.

$$3a_{1}=\frac{39}{22}$$

Whakawehea ngā taha e rua ki te 3.

$$a_{1}=\frac{13}{22}$$

Kua oti te pūnaha te whakatau.

$$a_{1}=\frac{13}{22},d=\frac{7}{66}$$