Whakaoti i te {c}{x+y=50}{300x+200y=11500}

$$\left. \begin{array} { c } { x + y = 50 } \\ { 300 x + 200 y = 11500 } \end{array} \right.$$

Whakaoti mō x, y

$$x=15$$

$$y=35$$

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.

$$x+y=50,300x+200y=11500$$

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

$$x+y=50$$

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

$$x=-y+50$$

Whakakapia te -y+50 mō te x ki tērā atu whārite, 300x+200y=11500.

$$300\left(-y+50\right)+200y=11500$$

Whakareatia 300 ki te -y+50.

$$-300y+15000+200y=11500$$

Tāpiri -300y ki te 200y.

$$-100y+15000=11500$$

Me tango 15000 mai i ngā taha e rua o te whārite.

$$-100y=-3500$$

Whakawehea ngā taha e rua ki te -100.

$$y=35$$

Whakaurua te 35 mō y ki x=-y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-35+50$$

Tāpiri 50 ki te -35.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=35$$

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.

$$x+y=50,300x+200y=11500$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\300&200\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\11500\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\300&200\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\300&200\end{matrix}\right))\left(\begin{matrix}1&1\\300&200\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\300&200\end{matrix}\right))\left(\begin{matrix}50\\11500\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\300&200\end{matrix}\right))\left(\begin{matrix}50\\11500\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\300&200\end{matrix}\right))\left(\begin{matrix}50\\11500\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{200-300}&-\frac{1}{200-300}\\-\frac{300}{200-300}&\frac{1}{200-300}\end{matrix}\right)\left(\begin{matrix}50\\11500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2&\frac{1}{100}\\3&-\frac{1}{100}\end{matrix}\right)\left(\begin{matrix}50\\11500\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\times 50+\frac{1}{100}\times 11500\\3\times 50-\frac{1}{100}\times 11500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\35\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=15,y=35$$

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.

$$x+y=50,300x+200y=11500$$

Kia ōrite ai a x me 300x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 300 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$300x+300y=300\times 50,300x+200y=11500$$

Whakarūnātia.

$$300x+300y=15000,300x+200y=11500$$

Me tango 300x+200y=11500 mai i 300x+300y=15000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$300x-300x+300y-200y=15000-11500$$

Tāpiri 300x ki te -300x. Ka whakakore atu ngā kupu 300x me -300x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$300y-200y=15000-11500$$

Tāpiri 300y ki te -200y.

$$100y=15000-11500$$

Tāpiri 15000 ki te -11500.

$$100y=3500$$

Whakawehea ngā taha e rua ki te 100.

$$y=35$$

Whakaurua te 35 mō y ki 300x+200y=11500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$300x+200\times 35=11500$$

Whakareatia 200 ki te 35.

$$300x+7000=11500$$

Me tango 7000 mai i ngā taha e rua o te whārite.

$$300x=4500$$

Whakawehea ngā taha e rua ki te 300.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=35$$