Whakaoti i te {c}{x+y=500}{50x+80y=28000}

$$\left. \begin{array} { c } { x + y = 500 } \\ { 50 x + 80 y = 28000 } \end{array} \right.$$

Whakaoti mō x, y

$$x=400$$

$$y=100$$

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.

$$x+y=500,50x+80y=28000$$

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

$$x+y=500$$

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

$$x=-y+500$$

Whakakapia te -y+500 mō te x ki tērā atu whārite, 50x+80y=28000.

$$50\left(-y+500\right)+80y=28000$$

Whakareatia 50 ki te -y+500.

$$-50y+25000+80y=28000$$

Tāpiri -50y ki te 80y.

$$30y+25000=28000$$

Me tango 25000 mai i ngā taha e rua o te whārite.

$$30y=3000$$

Whakawehea ngā taha e rua ki te 30.

$$y=100$$

Whakaurua te 100 mō y ki x=-y+500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-100+500$$

Tāpiri 500 ki te -100.

$$x=400$$

Kua oti te pūnaha te whakatau.

$$x=400,y=100$$

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.

$$x+y=500,50x+80y=28000$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\50&80\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}500\\28000\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\50&80\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\50&80\end{matrix}\right))\left(\begin{matrix}1&1\\50&80\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\50&80\end{matrix}\right))\left(\begin{matrix}500\\28000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\50&80\end{matrix}\right))\left(\begin{matrix}500\\28000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\50&80\end{matrix}\right))\left(\begin{matrix}500\\28000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{80}{80-50}&-\frac{1}{80-50}\\-\frac{50}{80-50}&\frac{1}{80-50}\end{matrix}\right)\left(\begin{matrix}500\\28000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}&-\frac{1}{30}\\-\frac{5}{3}&\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}500\\28000\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\times 500-\frac{1}{30}\times 28000\\-\frac{5}{3}\times 500+\frac{1}{30}\times 28000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}400\\100\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=400,y=100$$

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.

$$x+y=500,50x+80y=28000$$

Kia ōrite ai a x me 50x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 50 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$50x+50y=50\times 500,50x+80y=28000$$

Whakarūnātia.

$$50x+50y=25000,50x+80y=28000$$

Me tango 50x+80y=28000 mai i 50x+50y=25000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-50x+50y-80y=25000-28000$$

Tāpiri 50x ki te -50x. Ka whakakore atu ngā kupu 50x me -50x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$50y-80y=25000-28000$$

Tāpiri 50y ki te -80y.

$$-30y=25000-28000$$

Tāpiri 25000 ki te -28000.

$$-30y=-3000$$

Whakawehea ngā taha e rua ki te -30.

$$y=100$$

Whakaurua te 100 mō y ki 50x+80y=28000. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$50x+80\times 100=28000$$

Whakareatia 80 ki te 100.

$$50x+8000=28000$$

Me tango 8000 mai i ngā taha e rua o te whārite.

$$50x=20000$$

Whakawehea ngā taha e rua ki te 50.

$$x=400$$

Kua oti te pūnaha te whakatau.

$$x=400,y=100$$