Whakaoti i te {l}{x+y=50}{10x+20y=500}

$$\left\{ \begin{array} { l } { x + y = 50 } \\ { 10 x + 20 y = 500 } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=0$$

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.

$$x+y=50,10x+20y=500$$

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, 10x+20y=500.

$$10\left(-y+50\right)+20y=500$$

Whakareatia 10 ki te -y+50.

$$-10y+500+20y=500$$

Tāpiri -10y ki te 20y.

$$10y+500=500$$

Me tango 500 mai i ngā taha e rua o te whārite.

$$10y=0$$

Whakawehea ngā taha e rua ki te 10.

$$y=0$$

Whakaurua te 0 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.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=0$$

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.

$$x+y=50,10x+20y=500$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\10&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\500\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\10&20\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\10&20\end{matrix}\right))\left(\begin{matrix}1&1\\10&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\10&20\end{matrix}\right))\left(\begin{matrix}50\\500\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\10&20\end{matrix}\right))\left(\begin{matrix}50\\500\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\10&20\end{matrix}\right))\left(\begin{matrix}50\\500\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{20-10}&-\frac{1}{20-10}\\-\frac{10}{20-10}&\frac{1}{20-10}\end{matrix}\right)\left(\begin{matrix}50\\500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-\frac{1}{10}\\-1&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}50\\500\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 50-\frac{1}{10}\times 500\\-50+\frac{1}{10}\times 500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=0$$

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.

$$x+y=50,10x+20y=500$$

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

$$10x+10y=10\times 50,10x+20y=500$$

Whakarūnātia.

$$10x+10y=500,10x+20y=500$$

Me tango 10x+20y=500 mai i 10x+10y=500 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+10y-20y=500-500$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-20y=500-500$$

Tāpiri 10y ki te -20y.

$$-10y=500-500$$

Tāpiri 500 ki te -500.

$$-10y=0$$

Whakawehea ngā taha e rua ki te -10.

$$y=0$$

Whakaurua te 0 mō y ki 10x+20y=500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$10x=500$$

Whakawehea ngā taha e rua ki te 10.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=0$$