Whakaoti i te {c}{3x+2y=32}{365x+226y=267.6}

$$\left. \begin{array} { c } { 3 x + 2 y = 32 } \\ { 365 x + 226 y = 267.6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{8371}{65} = -128\frac{51}{65} \approx -128.784615385$$

$$y = \frac{27193}{130} = 209\frac{23}{130} \approx 209.176923077$$

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.

$$3x+2y=32,365x+226y=267.6$$

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

$$3x+2y=32$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y+32$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+32\right)$$

Whakareatia \frac{1}{3} ki te -2y+32.

$$x=-\frac{2}{3}y+\frac{32}{3}$$

Whakakapia te \frac{-2y+32}{3} mō te x ki tērā atu whārite, 365x+226y=267.6.

$$365\left(-\frac{2}{3}y+\frac{32}{3}\right)+226y=267.6$$

Whakareatia 365 ki te \frac{-2y+32}{3}.

$$-\frac{730}{3}y+\frac{11680}{3}+226y=267.6$$

Tāpiri -\frac{730y}{3} ki te 226y.

$$-\frac{52}{3}y+\frac{11680}{3}=267.6$$

Me tango \frac{11680}{3} mai i ngā taha e rua o te whārite.

$$-\frac{52}{3}y=-\frac{54386}{15}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{52}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{27193}{130}$$

Whakaurua te \frac{27193}{130} mō y ki x=-\frac{2}{3}y+\frac{32}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{27193}{130}+\frac{32}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{27193}{130} 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.

$$x=-\frac{27193}{195}+\frac{32}{3}$$

Tāpiri \frac{32}{3} ki te -\frac{27193}{195} 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.

$$x=-\frac{8371}{65}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{8371}{65},y=\frac{27193}{130}$$

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.

$$3x+2y=32,365x+226y=267.6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\365&226\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}32\\267.6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\365&226\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}3&2\\365&226\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}32\\267.6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}32\\267.6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}32\\267.6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{226}{3\times 226-2\times 365}&-\frac{2}{3\times 226-2\times 365}\\-\frac{365}{3\times 226-2\times 365}&\frac{3}{3\times 226-2\times 365}\end{matrix}\right)\left(\begin{matrix}32\\267.6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{113}{26}&\frac{1}{26}\\\frac{365}{52}&-\frac{3}{52}\end{matrix}\right)\left(\begin{matrix}32\\267.6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{113}{26}\times 32+\frac{1}{26}\times 267.6\\\frac{365}{52}\times 32-\frac{3}{52}\times 267.6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8371}{65}\\\frac{27193}{130}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{8371}{65},y=\frac{27193}{130}$$

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.

$$3x+2y=32,365x+226y=267.6$$

Kia ōrite ai a 3x me 365x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 365 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$365\times 3x+365\times 2y=365\times 32,3\times 365x+3\times 226y=3\times 267.6$$

Whakarūnātia.

$$1095x+730y=11680,1095x+678y=802.8$$

Me tango 1095x+678y=802.8 mai i 1095x+730y=11680 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$1095x-1095x+730y-678y=11680-802.8$$

Tāpiri 1095x ki te -1095x. Ka whakakore atu ngā kupu 1095x me -1095x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$730y-678y=11680-802.8$$

Tāpiri 730y ki te -678y.

$$52y=11680-802.8$$

Tāpiri 11680 ki te -802.8.

$$52y=10877.2$$

Whakawehea ngā taha e rua ki te 52.

$$y=\frac{27193}{130}$$

Whakaurua te \frac{27193}{130} mō y ki 365x+226y=267.6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$365x+226\times \frac{27193}{130}=267.6$$

Whakareatia 226 ki te \frac{27193}{130}.

$$365x+\frac{3072809}{65}=267.6$$

Me tango \frac{3072809}{65} mai i ngā taha e rua o te whārite.

$$365x=-\frac{611083}{13}$$

Whakawehea ngā taha e rua ki te 365.

$$x=-\frac{8371}{65}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{8371}{65},y=\frac{27193}{130}$$