Whakaoti i te {l}{15x+12y=1950}{7x+16y=1950}

$$\left\{ \begin{array} { l } { 15 x + 12 y = 1950 } \\ { 7 x + 16 y = 1950 } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=100$$

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.

$$15x+12y=1950,7x+16y=1950$$

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

$$15x+12y=1950$$

Me tango 12y mai i ngā taha e rua o te whārite.

$$15x=-12y+1950$$

Whakawehea ngā taha e rua ki te 15.

$$x=\frac{1}{15}\left(-12y+1950\right)$$

Whakareatia \frac{1}{15} ki te -12y+1950.

$$x=-\frac{4}{5}y+130$$

Whakakapia te -\frac{4y}{5}+130 mō te x ki tērā atu whārite, 7x+16y=1950.

$$7\left(-\frac{4}{5}y+130\right)+16y=1950$$

Whakareatia 7 ki te -\frac{4y}{5}+130.

$$-\frac{28}{5}y+910+16y=1950$$

Tāpiri -\frac{28y}{5} ki te 16y.

$$\frac{52}{5}y+910=1950$$

Me tango 910 mai i ngā taha e rua o te whārite.

$$\frac{52}{5}y=1040$$

Whakawehea ngā taha e rua o te whārite ki te \frac{52}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=100$$

Whakaurua te 100 mō y ki x=-\frac{4}{5}y+130. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{5}\times 100+130$$

Whakareatia -\frac{4}{5} ki te 100.

$$x=-80+130$$

Tāpiri 130 ki te -80.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=100$$

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.

$$15x+12y=1950,7x+16y=1950$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}15&12\\7&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1950\\1950\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}15&12\\7&16\end{matrix}\right).

$$inverse(\left(\begin{matrix}15&12\\7&16\end{matrix}\right))\left(\begin{matrix}15&12\\7&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&12\\7&16\end{matrix}\right))\left(\begin{matrix}1950\\1950\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&12\\7&16\end{matrix}\right))\left(\begin{matrix}1950\\1950\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&12\\7&16\end{matrix}\right))\left(\begin{matrix}1950\\1950\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{15\times 16-12\times 7}&-\frac{12}{15\times 16-12\times 7}\\-\frac{7}{15\times 16-12\times 7}&\frac{15}{15\times 16-12\times 7}\end{matrix}\right)\left(\begin{matrix}1950\\1950\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{39}&-\frac{1}{13}\\-\frac{7}{156}&\frac{5}{52}\end{matrix}\right)\left(\begin{matrix}1950\\1950\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{39}\times 1950-\frac{1}{13}\times 1950\\-\frac{7}{156}\times 1950+\frac{5}{52}\times 1950\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\100\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=100$$

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.

$$15x+12y=1950,7x+16y=1950$$

Kia ōrite ai a 15x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 15.

$$7\times 15x+7\times 12y=7\times 1950,15\times 7x+15\times 16y=15\times 1950$$

Whakarūnātia.

$$105x+84y=13650,105x+240y=29250$$

Me tango 105x+240y=29250 mai i 105x+84y=13650 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$105x-105x+84y-240y=13650-29250$$

Tāpiri 105x ki te -105x. Ka whakakore atu ngā kupu 105x me -105x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$84y-240y=13650-29250$$

Tāpiri 84y ki te -240y.

$$-156y=13650-29250$$

Tāpiri 13650 ki te -29250.

$$-156y=-15600$$

Whakawehea ngā taha e rua ki te -156.

$$y=100$$

Whakaurua te 100 mō y ki 7x+16y=1950. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+16\times 100=1950$$

Whakareatia 16 ki te 100.

$$7x+1600=1950$$

Me tango 1600 mai i ngā taha e rua o te whārite.

$$7x=350$$

Whakawehea ngā taha e rua ki te 7.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=100$$