Whakaoti i te {l}{-10x+20y=460}{30x+60y=1620}

$$\left. \begin{array} { l } { - 10 x + 20 y = 460 } \\ { 30 x + 60 y = 1620 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=25$$

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.

$$-10x+20y=460,30x+60y=1620$$

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

$$-10x+20y=460$$

Me tango 20y mai i ngā taha e rua o te whārite.

$$-10x=-20y+460$$

Whakawehea ngā taha e rua ki te -10.

$$x=-\frac{1}{10}\left(-20y+460\right)$$

Whakareatia -\frac{1}{10} ki te -20y+460.

$$x=2y-46$$

Whakakapia te -46+2y mō te x ki tērā atu whārite, 30x+60y=1620.

$$30\left(2y-46\right)+60y=1620$$

Whakareatia 30 ki te -46+2y.

$$60y-1380+60y=1620$$

Tāpiri 60y ki te 60y.

$$120y-1380=1620$$

Me tāpiri 1380 ki ngā taha e rua o te whārite.

$$120y=3000$$

Whakawehea ngā taha e rua ki te 120.

$$y=25$$

Whakaurua te 25 mō y ki x=2y-46. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\times 25-46$$

Whakareatia 2 ki te 25.

$$x=50-46$$

Tāpiri -46 ki te 50.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=25$$

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.

$$-10x+20y=460,30x+60y=1620$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-10&20\\30&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}460\\1620\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-10&20\\30&60\end{matrix}\right).

$$inverse(\left(\begin{matrix}-10&20\\30&60\end{matrix}\right))\left(\begin{matrix}-10&20\\30&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&20\\30&60\end{matrix}\right))\left(\begin{matrix}460\\1620\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&20\\30&60\end{matrix}\right))\left(\begin{matrix}460\\1620\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&20\\30&60\end{matrix}\right))\left(\begin{matrix}460\\1620\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{60}{-10\times 60-20\times 30}&-\frac{20}{-10\times 60-20\times 30}\\-\frac{30}{-10\times 60-20\times 30}&-\frac{10}{-10\times 60-20\times 30}\end{matrix}\right)\left(\begin{matrix}460\\1620\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{20}&\frac{1}{60}\\\frac{1}{40}&\frac{1}{120}\end{matrix}\right)\left(\begin{matrix}460\\1620\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{20}\times 460+\frac{1}{60}\times 1620\\\frac{1}{40}\times 460+\frac{1}{120}\times 1620\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\25\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=25$$

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.

$$-10x+20y=460,30x+60y=1620$$

Kia ōrite ai a -10x me 30x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 30 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -10.

$$30\left(-10\right)x+30\times 20y=30\times 460,-10\times 30x-10\times 60y=-10\times 1620$$

Whakarūnātia.

$$-300x+600y=13800,-300x-600y=-16200$$

Me tango -300x-600y=-16200 mai i -300x+600y=13800 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-300x+300x+600y+600y=13800+16200$$

Tāpiri -300x ki te 300x. Ka whakakore atu ngā kupu -300x me 300x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$600y+600y=13800+16200$$

Tāpiri 600y ki te 600y.

$$1200y=13800+16200$$

Tāpiri 13800 ki te 16200.

$$1200y=30000$$

Whakawehea ngā taha e rua ki te 1200.

$$y=25$$

Whakaurua te 25 mō y ki 30x+60y=1620. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$30x+60\times 25=1620$$

Whakareatia 60 ki te 25.

$$30x+1500=1620$$

Me tango 1500 mai i ngā taha e rua o te whārite.

$$30x=120$$

Whakawehea ngā taha e rua ki te 30.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=25$$