Whakaoti i te {l}{3A-13c=-255}{31A-6c=-180}

$$\left. \begin{array} { l } { 3 A - 13 c = - 255 } \\ { 31 A - 6 c = - 180 } \end{array} \right.$$

Whakaoti mō A, c

$$A = -\frac{162}{77} = -2\frac{8}{77} \approx -2.103896104$$

$$c = \frac{1473}{77} = 19\frac{10}{77} \approx 19.12987013$$

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.

$$3A-13c=-255,31A-6c=-180$$

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

$$3A-13c=-255$$

Me tāpiri 13c ki ngā taha e rua o te whārite.

$$3A=13c-255$$

Whakawehea ngā taha e rua ki te 3.

$$A=\frac{1}{3}\left(13c-255\right)$$

Whakareatia \frac{1}{3} ki te 13c-255.

$$A=\frac{13}{3}c-85$$

Whakakapia te \frac{13c}{3}-85 mō te A ki tērā atu whārite, 31A-6c=-180.

$$31\left(\frac{13}{3}c-85\right)-6c=-180$$

Whakareatia 31 ki te \frac{13c}{3}-85.

$$\frac{403}{3}c-2635-6c=-180$$

Tāpiri \frac{403c}{3} ki te -6c.

$$\frac{385}{3}c-2635=-180$$

Me tāpiri 2635 ki ngā taha e rua o te whārite.

$$\frac{385}{3}c=2455$$

Whakawehea ngā taha e rua o te whārite ki te \frac{385}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$c=\frac{1473}{77}$$

Whakaurua te \frac{1473}{77} mō c ki A=\frac{13}{3}c-85. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$A=\frac{13}{3}\times \frac{1473}{77}-85$$

Whakareatia \frac{13}{3} ki te \frac{1473}{77} 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.

$$A=\frac{6383}{77}-85$$

Tāpiri -85 ki te \frac{6383}{77}.

$$A=-\frac{162}{77}$$

Kua oti te pūnaha te whakatau.

$$A=-\frac{162}{77},c=\frac{1473}{77}$$

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.

$$3A-13c=-255,31A-6c=-180$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-13\\31&-6\end{matrix}\right)\left(\begin{matrix}A\\c\end{matrix}\right)=\left(\begin{matrix}-255\\-180\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-13\\31&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-13\\31&-6\end{matrix}\right))\left(\begin{matrix}3&-13\\31&-6\end{matrix}\right)\left(\begin{matrix}A\\c\end{matrix}\right)=inverse(\left(\begin{matrix}3&-13\\31&-6\end{matrix}\right))\left(\begin{matrix}-255\\-180\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}A\\c\end{matrix}\right)=inverse(\left(\begin{matrix}3&-13\\31&-6\end{matrix}\right))\left(\begin{matrix}-255\\-180\end{matrix}\right)$$

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

$$\left(\begin{matrix}A\\c\end{matrix}\right)=inverse(\left(\begin{matrix}3&-13\\31&-6\end{matrix}\right))\left(\begin{matrix}-255\\-180\end{matrix}\right)$$

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.

$$\left(\begin{matrix}A\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{3\left(-6\right)-\left(-13\times 31\right)}&-\frac{-13}{3\left(-6\right)-\left(-13\times 31\right)}\\-\frac{31}{3\left(-6\right)-\left(-13\times 31\right)}&\frac{3}{3\left(-6\right)-\left(-13\times 31\right)}\end{matrix}\right)\left(\begin{matrix}-255\\-180\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{385}&\frac{13}{385}\\-\frac{31}{385}&\frac{3}{385}\end{matrix}\right)\left(\begin{matrix}-255\\-180\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}A\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{385}\left(-255\right)+\frac{13}{385}\left(-180\right)\\-\frac{31}{385}\left(-255\right)+\frac{3}{385}\left(-180\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{162}{77}\\\frac{1473}{77}\end{matrix}\right)$$

Tangohia ngā huānga poukapa A me c.

$$A=-\frac{162}{77},c=\frac{1473}{77}$$

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.

$$3A-13c=-255,31A-6c=-180$$

Kia ōrite ai a 3A me 31A, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 31 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$31\times 3A+31\left(-13\right)c=31\left(-255\right),3\times 31A+3\left(-6\right)c=3\left(-180\right)$$

Whakarūnātia.

$$93A-403c=-7905,93A-18c=-540$$

Me tango 93A-18c=-540 mai i 93A-403c=-7905 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$93A-93A-403c+18c=-7905+540$$

Tāpiri 93A ki te -93A. Ka whakakore atu ngā kupu 93A me -93A, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-403c+18c=-7905+540$$

Tāpiri -403c ki te 18c.

$$-385c=-7905+540$$

Tāpiri -7905 ki te 540.

$$-385c=-7365$$

Whakawehea ngā taha e rua ki te -385.

$$c=\frac{1473}{77}$$

Whakaurua te \frac{1473}{77} mō c ki 31A-6c=-180. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$31A-6\times \frac{1473}{77}=-180$$

Whakareatia -6 ki te \frac{1473}{77}.

$$31A-\frac{8838}{77}=-180$$

Me tāpiri \frac{8838}{77} ki ngā taha e rua o te whārite.

$$31A=-\frac{5022}{77}$$

Whakawehea ngā taha e rua ki te 31.

$$A=-\frac{162}{77}$$

Kua oti te pūnaha te whakatau.

$$A=-\frac{162}{77},c=\frac{1473}{77}$$