Whakaoti i te {l}{x+y=100}{60x+70y=6300}

$$\left. \begin{array} { l } { x + y = 100 } \\ { 60 x + 70 y = 6300 } \end{array} \right.$$

Whakaoti mō x, y

$$x=70$$

$$y=30$$

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.

$$x+y=100,60x+70y=6300$$

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

$$x+y=100$$

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

$$x=-y+100$$

Whakakapia te -y+100 mō te x ki tērā atu whārite, 60x+70y=6300.

$$60\left(-y+100\right)+70y=6300$$

Whakareatia 60 ki te -y+100.

$$-60y+6000+70y=6300$$

Tāpiri -60y ki te 70y.

$$10y+6000=6300$$

Me tango 6000 mai i ngā taha e rua o te whārite.

$$10y=300$$

Whakawehea ngā taha e rua ki te 10.

$$y=30$$

Whakaurua te 30 mō y ki x=-y+100. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-30+100$$

Tāpiri 100 ki te -30.

$$x=70$$

Kua oti te pūnaha te whakatau.

$$x=70,y=30$$

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.

$$x+y=100,60x+70y=6300$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\60&70\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\6300\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\60&70\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\60&70\end{matrix}\right))\left(\begin{matrix}1&1\\60&70\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\60&70\end{matrix}\right))\left(\begin{matrix}100\\6300\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\60&70\end{matrix}\right))\left(\begin{matrix}100\\6300\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\60&70\end{matrix}\right))\left(\begin{matrix}100\\6300\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{70}{70-60}&-\frac{1}{70-60}\\-\frac{60}{70-60}&\frac{1}{70-60}\end{matrix}\right)\left(\begin{matrix}100\\6300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7&-\frac{1}{10}\\-6&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}100\\6300\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\times 100-\frac{1}{10}\times 6300\\-6\times 100+\frac{1}{10}\times 6300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}70\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=70,y=30$$

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.

$$x+y=100,60x+70y=6300$$

Kia ōrite ai a x me 60x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 60 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$60x+60y=60\times 100,60x+70y=6300$$

Whakarūnātia.

$$60x+60y=6000,60x+70y=6300$$

Me tango 60x+70y=6300 mai i 60x+60y=6000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$60x-60x+60y-70y=6000-6300$$

Tāpiri 60x ki te -60x. Ka whakakore atu ngā kupu 60x me -60x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$60y-70y=6000-6300$$

Tāpiri 60y ki te -70y.

$$-10y=6000-6300$$

Tāpiri 6000 ki te -6300.

$$-10y=-300$$

Whakawehea ngā taha e rua ki te -10.

$$y=30$$

Whakaurua te 30 mō y ki 60x+70y=6300. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$60x+70\times 30=6300$$

Whakareatia 70 ki te 30.

$$60x+2100=6300$$

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

$$60x=4200$$

Whakawehea ngā taha e rua ki te 60.

$$x=70$$

Kua oti te pūnaha te whakatau.

$$x=70,y=30$$