Whakaoti i te {l}{18=1200x+1600y}{17=600x+2400y}

$$\left. \begin{array} { l } { 18 = 1200 x + 1600 y } \\ { 17 = 600 x + 2400 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{120}\approx 0.008333333$$

$$y=\frac{1}{200}=0.005$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1200x+1600y=18$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$600x+2400y=17$$

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.

$$1200x+1600y=18,600x+2400y=17$$

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

$$1200x+1600y=18$$

Me tango 1600y mai i ngā taha e rua o te whārite.

$$1200x=-1600y+18$$

Whakawehea ngā taha e rua ki te 1200.

$$x=\frac{1}{1200}\left(-1600y+18\right)$$

Whakareatia \frac{1}{1200} ki te -1600y+18.

$$x=-\frac{4}{3}y+\frac{3}{200}$$

Whakakapia te -\frac{4y}{3}+\frac{3}{200} mō te x ki tērā atu whārite, 600x+2400y=17.

$$600\left(-\frac{4}{3}y+\frac{3}{200}\right)+2400y=17$$

Whakareatia 600 ki te -\frac{4y}{3}+\frac{3}{200}.

$$-800y+9+2400y=17$$

Tāpiri -800y ki te 2400y.

$$1600y+9=17$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$1600y=8$$

Whakawehea ngā taha e rua ki te 1600.

$$y=\frac{1}{200}$$

Whakaurua te \frac{1}{200} mō y ki x=-\frac{4}{3}y+\frac{3}{200}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\times \frac{1}{200}+\frac{3}{200}$$

Whakareatia -\frac{4}{3} ki te \frac{1}{200} 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.

$$x=-\frac{1}{150}+\frac{3}{200}$$

Tāpiri \frac{3}{200} ki te -\frac{1}{150} 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.

$$x=\frac{1}{120}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{120},y=\frac{1}{200}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1200x+1600y=18$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$600x+2400y=17$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$1200x+1600y=18,600x+2400y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right).

$$inverse(\left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right))\left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right))\left(\begin{matrix}18\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right))\left(\begin{matrix}18\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1200&1600\\600&2400\end{matrix}\right))\left(\begin{matrix}18\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2400}{1200\times 2400-1600\times 600}&-\frac{1600}{1200\times 2400-1600\times 600}\\-\frac{600}{1200\times 2400-1600\times 600}&\frac{1200}{1200\times 2400-1600\times 600}\end{matrix}\right)\left(\begin{matrix}18\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{800}&-\frac{1}{1200}\\-\frac{1}{3200}&\frac{1}{1600}\end{matrix}\right)\left(\begin{matrix}18\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{800}\times 18-\frac{1}{1200}\times 17\\-\frac{1}{3200}\times 18+\frac{1}{1600}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{120}\\\frac{1}{200}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{120},y=\frac{1}{200}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1200x+1600y=18$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$600x+2400y=17$$

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.

$$1200x+1600y=18,600x+2400y=17$$

Kia ōrite ai a 1200x me 600x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 600 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1200.

$$600\times 1200x+600\times 1600y=600\times 18,1200\times 600x+1200\times 2400y=1200\times 17$$

Whakarūnātia.

$$720000x+960000y=10800,720000x+2880000y=20400$$

Me tango 720000x+2880000y=20400 mai i 720000x+960000y=10800 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$720000x-720000x+960000y-2880000y=10800-20400$$

Tāpiri 720000x ki te -720000x. Ka whakakore atu ngā kupu 720000x me -720000x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$960000y-2880000y=10800-20400$$

Tāpiri 960000y ki te -2880000y.

$$-1920000y=10800-20400$$

Tāpiri 10800 ki te -20400.

$$-1920000y=-9600$$

Whakawehea ngā taha e rua ki te -1920000.

$$y=\frac{1}{200}$$

Whakaurua te \frac{1}{200} mō y ki 600x+2400y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$600x+2400\times \frac{1}{200}=17$$

Whakareatia 2400 ki te \frac{1}{200}.

$$600x+12=17$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$600x=5$$

Whakawehea ngā taha e rua ki te 600.

$$x=\frac{1}{120}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{120},y=\frac{1}{200}$$