Whakaoti i te {l}{X+y=2}{60X=200y}

$$\left. \begin{array} { l } { X + y = 2 } \\ { 60 X = 200 y } \end{array} \right.$$

Whakaoti mō X, y

$$y=\frac{6}{13}\approx 0.461538462$$

$$X = \frac{20}{13} = 1\frac{7}{13} \approx 1.538461538$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 200y mai i ngā taha e rua.

$$60X-200y=0$$

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.

$$X+y=2,60X-200y=0$$

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

$$X+y=2$$

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

$$X=-y+2$$

Whakakapia te -y+2 mō te X ki tērā atu whārite, 60X-200y=0.

$$60\left(-y+2\right)-200y=0$$

Whakareatia 60 ki te -y+2.

$$-60y+120-200y=0$$

Tāpiri -60y ki te -200y.

$$-260y+120=0$$

Me tango 120 mai i ngā taha e rua o te whārite.

$$-260y=-120$$

Whakawehea ngā taha e rua ki te -260.

$$y=\frac{6}{13}$$

Whakaurua te \frac{6}{13} mō y ki X=-y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō X hāngai tonu.

$$X=-\frac{6}{13}+2$$

Tāpiri 2 ki te -\frac{6}{13}.

$$X=\frac{20}{13}$$

Kua oti te pūnaha te whakatau.

$$X=\frac{20}{13},y=\frac{6}{13}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 200y mai i ngā taha e rua.

$$60X-200y=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$X+y=2,60X-200y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\60&-200\end{matrix}\right)\left(\begin{matrix}X\\y\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\60&-200\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\60&-200\end{matrix}\right))\left(\begin{matrix}1&1\\60&-200\end{matrix}\right)\left(\begin{matrix}X\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\60&-200\end{matrix}\right))\left(\begin{matrix}2\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}X\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\60&-200\end{matrix}\right))\left(\begin{matrix}2\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}X\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\60&-200\end{matrix}\right))\left(\begin{matrix}2\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}X\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{200}{-200-60}&-\frac{1}{-200-60}\\-\frac{60}{-200-60}&\frac{1}{-200-60}\end{matrix}\right)\left(\begin{matrix}2\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}X\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{13}&\frac{1}{260}\\\frac{3}{13}&-\frac{1}{260}\end{matrix}\right)\left(\begin{matrix}2\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}X\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{13}\times 2\\\frac{3}{13}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}X\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{13}\\\frac{6}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa X me y.

$$X=\frac{20}{13},y=\frac{6}{13}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 200y mai i ngā taha e rua.

$$60X-200y=0$$

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.

$$X+y=2,60X-200y=0$$

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.

$$60X+60y=60\times 2,60X-200y=0$$

Whakarūnātia.

$$60X+60y=120,60X-200y=0$$

Me tango 60X-200y=0 mai i 60X+60y=120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$60X-60X+60y+200y=120$$

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.

$$60y+200y=120$$

Tāpiri 60y ki te 200y.

$$260y=120$$

Whakawehea ngā taha e rua ki te 260.

$$y=\frac{6}{13}$$

Whakaurua te \frac{6}{13} mō y ki 60X-200y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō X hāngai tonu.

$$60X-200\times \frac{6}{13}=0$$

Whakareatia -200 ki te \frac{6}{13}.

$$60X-\frac{1200}{13}=0$$

Me tāpiri \frac{1200}{13} ki ngā taha e rua o te whārite.

$$60X=\frac{1200}{13}$$

Whakawehea ngā taha e rua ki te 60.

$$X=\frac{20}{13}$$

Kua oti te pūnaha te whakatau.

$$X=\frac{20}{13},y=\frac{6}{13}$$