Whakaoti i te {c}{1/10x+1/2y=1}{2x-10y=-20}

$$\left. \begin{array} { c } { \frac { 1 } { 10 } x + \frac { 1 } { 2 } y = 1 } \\ { 2 x - 10 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=2$$

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.

$$\frac{1}{10}x+\frac{1}{2}y=1,2x-10y=-20$$

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

$$\frac{1}{10}x+\frac{1}{2}y=1$$

Me tango \frac{y}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{10}x=-\frac{1}{2}y+1$$

Me whakarea ngā taha e rua ki te 10.

$$x=10\left(-\frac{1}{2}y+1\right)$$

Whakareatia 10 ki te -\frac{y}{2}+1.

$$x=-5y+10$$

Whakakapia te -5y+10 mō te x ki tērā atu whārite, 2x-10y=-20.

$$2\left(-5y+10\right)-10y=-20$$

Whakareatia 2 ki te -5y+10.

$$-10y+20-10y=-20$$

Tāpiri -10y ki te -10y.

$$-20y+20=-20$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-20y=-40$$

Whakawehea ngā taha e rua ki te -20.

$$y=2$$

Whakaurua te 2 mō y ki x=-5y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\times 2+10$$

Whakareatia -5 ki te 2.

$$x=-10+10$$

Tāpiri 10 ki te -10.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$

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.

$$\frac{1}{10}x+\frac{1}{2}y=1,2x-10y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right))\left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right))\left(\begin{matrix}1\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right))\left(\begin{matrix}1\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{10}&\frac{1}{2}\\2&-10\end{matrix}\right))\left(\begin{matrix}1\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{\frac{1}{10}\left(-10\right)-\frac{1}{2}\times 2}&-\frac{\frac{1}{2}}{\frac{1}{10}\left(-10\right)-\frac{1}{2}\times 2}\\-\frac{2}{\frac{1}{10}\left(-10\right)-\frac{1}{2}\times 2}&\frac{\frac{1}{10}}{\frac{1}{10}\left(-10\right)-\frac{1}{2}\times 2}\end{matrix}\right)\left(\begin{matrix}1\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&\frac{1}{4}\\1&-\frac{1}{20}\end{matrix}\right)\left(\begin{matrix}1\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5+\frac{1}{4}\left(-20\right)\\1-\frac{1}{20}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=2$$

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.

$$\frac{1}{10}x+\frac{1}{2}y=1,2x-10y=-20$$

Kia ōrite ai a \frac{x}{10} me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{10}.

$$2\times \frac{1}{10}x+2\times \frac{1}{2}y=2,\frac{1}{10}\times 2x+\frac{1}{10}\left(-10\right)y=\frac{1}{10}\left(-20\right)$$

Whakarūnātia.

$$\frac{1}{5}x+y=2,\frac{1}{5}x-y=-2$$

Me tango \frac{1}{5}x-y=-2 mai i \frac{1}{5}x+y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{5}x-\frac{1}{5}x+y+y=2+2$$

Tāpiri \frac{x}{5} ki te -\frac{x}{5}. Ka whakakore atu ngā kupu \frac{x}{5} me -\frac{x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y+y=2+2$$

Tāpiri y ki te y.

$$2y=2+2$$

Tāpiri 2 ki te 2.

$$2y=4$$

Whakawehea ngā taha e rua ki te 2.

$$y=2$$

Whakaurua te 2 mō y ki 2x-10y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-10\times 2=-20$$

Whakareatia -10 ki te 2.

$$2x-20=-20$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$2x=0$$

Whakawehea ngā taha e rua ki te 2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$