Whakaoti i te {l}{-15A+3B=21}{-15B-3A=-14}

$$\left. \begin{array} { l } { - 15 A + 3 B = 21 } \\ { - 15 B - 3 A = - 14 } \end{array} \right.$$

Whakaoti mō A, B

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

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

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.

$$-15A+3B=21,-3A-15B=-14$$

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

$$-15A+3B=21$$

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

$$-15A=-3B+21$$

Whakawehea ngā taha e rua ki te -15.

$$A=-\frac{1}{15}\left(-3B+21\right)$$

Whakareatia -\frac{1}{15} ki te -3B+21.

$$A=\frac{1}{5}B-\frac{7}{5}$$

Whakakapia te \frac{-7+B}{5} mō te A ki tērā atu whārite, -3A-15B=-14.

$$-3\left(\frac{1}{5}B-\frac{7}{5}\right)-15B=-14$$

Whakareatia -3 ki te \frac{-7+B}{5}.

$$-\frac{3}{5}B+\frac{21}{5}-15B=-14$$

Tāpiri -\frac{3B}{5} ki te -15B.

$$-\frac{78}{5}B+\frac{21}{5}=-14$$

Me tango \frac{21}{5} mai i ngā taha e rua o te whārite.

$$-\frac{78}{5}B=-\frac{91}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{78}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

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

$$A=\frac{1}{5}\times \frac{7}{6}-\frac{7}{5}$$

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.

$$A=\frac{7}{30}-\frac{7}{5}$$

Tāpiri -\frac{7}{5} 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.

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

Kua oti te pūnaha te whakatau.

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

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.

$$-15A+3B=21,-3A-15B=-14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-15&3\\-3&-15\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}21\\-14\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-15&3\\-3&-15\end{matrix}\right))\left(\begin{matrix}-15&3\\-3&-15\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}-15&3\\-3&-15\end{matrix}\right))\left(\begin{matrix}21\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}-15&3\\-3&-15\end{matrix}\right))\left(\begin{matrix}21\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}-15&3\\-3&-15\end{matrix}\right))\left(\begin{matrix}21\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{-15\left(-15\right)-3\left(-3\right)}&-\frac{3}{-15\left(-15\right)-3\left(-3\right)}\\-\frac{-3}{-15\left(-15\right)-3\left(-3\right)}&-\frac{15}{-15\left(-15\right)-3\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}21\\-14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{78}&-\frac{1}{78}\\\frac{1}{78}&-\frac{5}{78}\end{matrix}\right)\left(\begin{matrix}21\\-14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{78}\times 21-\frac{1}{78}\left(-14\right)\\\frac{1}{78}\times 21-\frac{5}{78}\left(-14\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa A me B.

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

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.

$$-15A+3B=21,-3A-15B=-14$$

Kia ōrite ai a -15A me -3A, 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 -15.

$$-3\left(-15\right)A-3\times 3B=-3\times 21,-15\left(-3\right)A-15\left(-15\right)B=-15\left(-14\right)$$

Whakarūnātia.

$$45A-9B=-63,45A+225B=210$$

Me tango 45A+225B=210 mai i 45A-9B=-63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$45A-45A-9B-225B=-63-210$$

Tāpiri 45A ki te -45A. Ka whakakore atu ngā kupu 45A me -45A, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9B-225B=-63-210$$

Tāpiri -9B ki te -225B.

$$-234B=-63-210$$

Tāpiri -63 ki te -210.

$$-234B=-273$$

Whakawehea ngā taha e rua ki te -234.

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

Whakaurua te \frac{7}{6} mō B ki -3A-15B=-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$-3A-15\times \frac{7}{6}=-14$$

Whakareatia -15 ki te \frac{7}{6}.

$$-3A-\frac{35}{2}=-14$$

Me tāpiri \frac{35}{2} ki ngā taha e rua o te whārite.

$$-3A=\frac{7}{2}$$

Whakawehea ngā taha e rua ki te -3.

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

Kua oti te pūnaha te whakatau.

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