Whakaoti i te {l}{-15B-3A=-14}{-5A+B=7}

$$\left. \begin{array} { l } { - 15 B - 3 A = - 14 } \\ { - 5 A + B = 7 } \end{array} \right.$$

Whakaoti mō B, A

$$B = \frac{7}{6} = 1\frac{1}{6} \approx 1.166666667$$

$$A = -\frac{7}{6} = -1\frac{1}{6} \approx -1.166666667$$

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.

$$-15B-3A=-14,B-5A=7$$

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

$$-15B-3A=-14$$

Me tāpiri 3A ki ngā taha e rua o te whārite.

$$-15B=3A-14$$

Whakawehea ngā taha e rua ki te -15.

$$B=-\frac{1}{15}\left(3A-14\right)$$

Whakareatia -\frac{1}{15} ki te 3A-14.

$$B=-\frac{1}{5}A+\frac{14}{15}$$

Whakakapia te -\frac{A}{5}+\frac{14}{15} mō te B ki tērā atu whārite, B-5A=7.

$$-\frac{1}{5}A+\frac{14}{15}-5A=7$$

Tāpiri -\frac{A}{5} ki te -5A.

$$-\frac{26}{5}A+\frac{14}{15}=7$$

Me tango \frac{14}{15} mai i ngā taha e rua o te whārite.

$$-\frac{26}{5}A=\frac{91}{15}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{26}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$A=-\frac{7}{6}$$

Whakaurua te -\frac{7}{6} mō A ki B=-\frac{1}{5}A+\frac{14}{15}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō B hāngai tonu.

$$B=-\frac{1}{5}\left(-\frac{7}{6}\right)+\frac{14}{15}$$

Whakareatia -\frac{1}{5} ki te -\frac{7}{6} 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.

$$B=\frac{7}{30}+\frac{14}{15}$$

Tāpiri \frac{14}{15} ki te \frac{7}{30} 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.

$$B=\frac{7}{6}$$

Kua oti te pūnaha te whakatau.

$$B=\frac{7}{6},A=-\frac{7}{6}$$

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.

$$-15B-3A=-14,B-5A=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right)\left(\begin{matrix}B\\A\end{matrix}\right)=\left(\begin{matrix}-14\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right))\left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right)\left(\begin{matrix}B\\A\end{matrix}\right)=inverse(\left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right))\left(\begin{matrix}-14\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}B\\A\end{matrix}\right)=inverse(\left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right))\left(\begin{matrix}-14\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}B\\A\end{matrix}\right)=inverse(\left(\begin{matrix}-15&-3\\1&-5\end{matrix}\right))\left(\begin{matrix}-14\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}B\\A\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-15\left(-5\right)-\left(-3\right)}&-\frac{-3}{-15\left(-5\right)-\left(-3\right)}\\-\frac{1}{-15\left(-5\right)-\left(-3\right)}&-\frac{15}{-15\left(-5\right)-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}-14\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}B\\A\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{78}&\frac{1}{26}\\-\frac{1}{78}&-\frac{5}{26}\end{matrix}\right)\left(\begin{matrix}-14\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}B\\A\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{78}\left(-14\right)+\frac{1}{26}\times 7\\-\frac{1}{78}\left(-14\right)-\frac{5}{26}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}B\\A\end{matrix}\right)=\left(\begin{matrix}\frac{7}{6}\\-\frac{7}{6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa B me A.

$$B=\frac{7}{6},A=-\frac{7}{6}$$

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.

$$-15B-3A=-14,B-5A=7$$

Kia ōrite ai a -15B me B, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -15.

$$-15B-3A=-14,-15B-15\left(-5\right)A=-15\times 7$$

Whakarūnātia.

$$-15B-3A=-14,-15B+75A=-105$$

Me tango -15B+75A=-105 mai i -15B-3A=-14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15B+15B-3A-75A=-14+105$$

Tāpiri -15B ki te 15B. Ka whakakore atu ngā kupu -15B me 15B, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3A-75A=-14+105$$

Tāpiri -3A ki te -75A.

$$-78A=-14+105$$

Tāpiri -14 ki te 105.

$$-78A=91$$

Whakawehea ngā taha e rua ki te -78.

$$A=-\frac{7}{6}$$

Whakaurua te -\frac{7}{6} mō A ki B-5A=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō B hāngai tonu.

$$B-5\left(-\frac{7}{6}\right)=7$$

Whakareatia -5 ki te -\frac{7}{6}.

$$B+\frac{35}{6}=7$$

Me tango \frac{35}{6} mai i ngā taha e rua o te whārite.

$$B=\frac{7}{6}$$

Kua oti te pūnaha te whakatau.

$$B=\frac{7}{6},A=-\frac{7}{6}$$