Whakaoti i te {l}{8x+20y=11400}{10x+30y=22500}

$$\left\{ \begin{array} { l } { 8 x + 20 y = 11400 } \\ { 10 x + 30 y = 22500 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2700$$

$$y=1650$$

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.

$$8x+20y=11400,10x+30y=22500$$

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

$$8x+20y=11400$$

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

$$8x=-20y+11400$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-20y+11400\right)$$

Whakareatia \frac{1}{8} ki te -20y+11400.

$$x=-\frac{5}{2}y+1425$$

Whakakapia te -\frac{5y}{2}+1425 mō te x ki tērā atu whārite, 10x+30y=22500.

$$10\left(-\frac{5}{2}y+1425\right)+30y=22500$$

Whakareatia 10 ki te -\frac{5y}{2}+1425.

$$-25y+14250+30y=22500$$

Tāpiri -25y ki te 30y.

$$5y+14250=22500$$

Me tango 14250 mai i ngā taha e rua o te whārite.

$$5y=8250$$

Whakawehea ngā taha e rua ki te 5.

$$y=1650$$

Whakaurua te 1650 mō y ki x=-\frac{5}{2}y+1425. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\times 1650+1425$$

Whakareatia -\frac{5}{2} ki te 1650.

$$x=-4125+1425$$

Tāpiri 1425 ki te -4125.

$$x=-2700$$

Kua oti te pūnaha te whakatau.

$$x=-2700,y=1650$$

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.

$$8x+20y=11400,10x+30y=22500$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&20\\10&30\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11400\\22500\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&20\\10&30\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&20\\10&30\end{matrix}\right))\left(\begin{matrix}8&20\\10&30\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&20\\10&30\end{matrix}\right))\left(\begin{matrix}11400\\22500\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&20\\10&30\end{matrix}\right))\left(\begin{matrix}11400\\22500\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&20\\10&30\end{matrix}\right))\left(\begin{matrix}11400\\22500\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{30}{8\times 30-20\times 10}&-\frac{20}{8\times 30-20\times 10}\\-\frac{10}{8\times 30-20\times 10}&\frac{8}{8\times 30-20\times 10}\end{matrix}\right)\left(\begin{matrix}11400\\22500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&-\frac{1}{2}\\-\frac{1}{4}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}11400\\22500\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 11400-\frac{1}{2}\times 22500\\-\frac{1}{4}\times 11400+\frac{1}{5}\times 22500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2700\\1650\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-2700,y=1650$$

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.

$$8x+20y=11400,10x+30y=22500$$

Kia ōrite ai a 8x me 10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$10\times 8x+10\times 20y=10\times 11400,8\times 10x+8\times 30y=8\times 22500$$

Whakarūnātia.

$$80x+200y=114000,80x+240y=180000$$

Me tango 80x+240y=180000 mai i 80x+200y=114000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$80x-80x+200y-240y=114000-180000$$

Tāpiri 80x ki te -80x. Ka whakakore atu ngā kupu 80x me -80x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$200y-240y=114000-180000$$

Tāpiri 200y ki te -240y.

$$-40y=114000-180000$$

Tāpiri 114000 ki te -180000.

$$-40y=-66000$$

Whakawehea ngā taha e rua ki te -40.

$$y=1650$$

Whakaurua te 1650 mō y ki 10x+30y=22500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$10x+30\times 1650=22500$$

Whakareatia 30 ki te 1650.

$$10x+49500=22500$$

Me tango 49500 mai i ngā taha e rua o te whārite.

$$10x=-27000$$

Whakawehea ngā taha e rua ki te 10.

$$x=-2700$$

Kua oti te pūnaha te whakatau.

$$x=-2700,y=1650$$