Whakaoti i te {l}{x+y=74}{40x+60y=3660}

$$\left. \begin{array} { l } { x + y = 74 } \\ { 40 x + 60 y = 3660 } \end{array} \right.$$

Whakaoti mō x, y

$$x=39$$

$$y=35$$

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.

$$x+y=74,40x+60y=3660$$

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

$$x+y=74$$

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

$$x=-y+74$$

Whakakapia te -y+74 mō te x ki tērā atu whārite, 40x+60y=3660.

$$40\left(-y+74\right)+60y=3660$$

Whakareatia 40 ki te -y+74.

$$-40y+2960+60y=3660$$

Tāpiri -40y ki te 60y.

$$20y+2960=3660$$

Me tango 2960 mai i ngā taha e rua o te whārite.

$$20y=700$$

Whakawehea ngā taha e rua ki te 20.

$$y=35$$

Whakaurua te 35 mō y ki x=-y+74. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-35+74$$

Tāpiri 74 ki te -35.

$$x=39$$

Kua oti te pūnaha te whakatau.

$$x=39,y=35$$

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.

$$x+y=74,40x+60y=3660$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\40&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}74\\3660\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\40&60\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\40&60\end{matrix}\right))\left(\begin{matrix}1&1\\40&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&60\end{matrix}\right))\left(\begin{matrix}74\\3660\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&60\end{matrix}\right))\left(\begin{matrix}74\\3660\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&60\end{matrix}\right))\left(\begin{matrix}74\\3660\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{60}{60-40}&-\frac{1}{60-40}\\-\frac{40}{60-40}&\frac{1}{60-40}\end{matrix}\right)\left(\begin{matrix}74\\3660\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&-\frac{1}{20}\\-2&\frac{1}{20}\end{matrix}\right)\left(\begin{matrix}74\\3660\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 74-\frac{1}{20}\times 3660\\-2\times 74+\frac{1}{20}\times 3660\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}39\\35\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=39,y=35$$

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.

$$x+y=74,40x+60y=3660$$

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.

$$40x+40y=40\times 74,40x+60y=3660$$

Whakarūnātia.

$$40x+40y=2960,40x+60y=3660$$

Me tango 40x+60y=3660 mai i 40x+40y=2960 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$40x-40x+40y-60y=2960-3660$$

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.

$$40y-60y=2960-3660$$

Tāpiri 40y ki te -60y.

$$-20y=2960-3660$$

Tāpiri 2960 ki te -3660.

$$-20y=-700$$

Whakawehea ngā taha e rua ki te -20.

$$y=35$$

Whakaurua te 35 mō y ki 40x+60y=3660. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$40x+60\times 35=3660$$

Whakareatia 60 ki te 35.

$$40x+2100=3660$$

Me tango 2100 mai i ngā taha e rua o te whārite.

$$40x=1560$$

Whakawehea ngā taha e rua ki te 40.

$$x=39$$

Kua oti te pūnaha te whakatau.

$$x=39,y=35$$