Whakaoti i te {c}{x+y=8}{40x+55y=410}

$$\left. \begin{array} { c } { x + y = 8 } \\ { 40 x + 55 y = 410 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=6$$

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.

$$x+y=8,40x+55y=410$$

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

$$x+y=8$$

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

$$x=-y+8$$

Whakakapia te -y+8 mō te x ki tērā atu whārite, 40x+55y=410.

$$40\left(-y+8\right)+55y=410$$

Whakareatia 40 ki te -y+8.

$$-40y+320+55y=410$$

Tāpiri -40y ki te 55y.

$$15y+320=410$$

Me tango 320 mai i ngā taha e rua o te whārite.

$$15y=90$$

Whakawehea ngā taha e rua ki te 15.

$$y=6$$

Whakaurua te 6 mō y ki x=-y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6+8$$

Tāpiri 8 ki te -6.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=6$$

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.

$$x+y=8,40x+55y=410$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\40&55\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\410\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\40&55\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\40&55\end{matrix}\right))\left(\begin{matrix}1&1\\40&55\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&55\end{matrix}\right))\left(\begin{matrix}8\\410\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&55\end{matrix}\right))\left(\begin{matrix}8\\410\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&55\end{matrix}\right))\left(\begin{matrix}8\\410\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{55}{55-40}&-\frac{1}{55-40}\\-\frac{40}{55-40}&\frac{1}{55-40}\end{matrix}\right)\left(\begin{matrix}8\\410\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{3}&-\frac{1}{15}\\-\frac{8}{3}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}8\\410\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{3}\times 8-\frac{1}{15}\times 410\\-\frac{8}{3}\times 8+\frac{1}{15}\times 410\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=6$$

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.

$$x+y=8,40x+55y=410$$

Kia ōrite ai a x me 40x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 40 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$40x+40y=40\times 8,40x+55y=410$$

Whakarūnātia.

$$40x+40y=320,40x+55y=410$$

Me tango 40x+55y=410 mai i 40x+40y=320 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$40x-40x+40y-55y=320-410$$

Tāpiri 40x ki te -40x. Ka whakakore atu ngā kupu 40x me -40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40y-55y=320-410$$

Tāpiri 40y ki te -55y.

$$-15y=320-410$$

Tāpiri 320 ki te -410.

$$-15y=-90$$

Whakawehea ngā taha e rua ki te -15.

$$y=6$$

Whakaurua te 6 mō y ki 40x+55y=410. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$40x+55\times 6=410$$

Whakareatia 55 ki te 6.

$$40x+330=410$$

Me tango 330 mai i ngā taha e rua o te whārite.

$$40x=80$$

Whakawehea ngā taha e rua ki te 40.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=6$$