Whakaoti i te {l}{x-y=10}{2x+2,5y=200}

$$\left\{ \begin{array} { l } { x - y = 10 } \\ { 2 x + 2,5 y = 200 } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=40$$

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.

$$x-y=10,2x+2.5y=200$$

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

$$x-y=10$$

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

$$x=y+10$$

Whakakapia te y+10 mō te x ki tērā atu whārite, 2x+2.5y=200.

$$2\left(y+10\right)+2.5y=200$$

Whakareatia 2 ki te y+10.

$$2y+20+2.5y=200$$

Tāpiri 2y ki te \frac{5y}{2}.

$$4.5y+20=200$$

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

$$4.5y=180$$

Whakawehea ngā taha e rua o te whārite ki te 4.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=40$$

Whakaurua te 40 mō y ki x=y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=40+10$$

Tāpiri 10 ki te 40.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=40$$

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.

$$x-y=10,2x+2.5y=200$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\200\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right))\left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right))\left(\begin{matrix}10\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right))\left(\begin{matrix}10\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2.5\end{matrix}\right))\left(\begin{matrix}10\\200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2.5}{2.5-\left(-2\right)}&-\frac{-1}{2.5-\left(-2\right)}\\-\frac{2}{2.5-\left(-2\right)}&\frac{1}{2.5-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}10\\200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{9}&\frac{2}{9}\\-\frac{4}{9}&\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}10\\200\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{9}\times 10+\frac{2}{9}\times 200\\-\frac{4}{9}\times 10+\frac{2}{9}\times 200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\40\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=40$$

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.

$$x-y=10,2x+2.5y=200$$

Kia ōrite ai a x 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 1.

$$2x+2\left(-1\right)y=2\times 10,2x+2.5y=200$$

Whakarūnātia.

$$2x-2y=20,2x+2.5y=200$$

Me tango 2x+2.5y=200 mai i 2x-2y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-2y-2.5y=20-200$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-2.5y=20-200$$

Tāpiri -2y ki te -\frac{5y}{2}.

$$-4.5y=20-200$$

Tāpiri 20 ki te -200.

$$-4.5y=-180$$

Whakawehea ngā taha e rua o te whārite ki te -4.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=40$$

Whakaurua te 40 mō y ki 2x+2.5y=200. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+2.5\times 40=200$$

Whakareatia 2.5 ki te 40.

$$2x+100=200$$

Me tango 100 mai i ngā taha e rua o te whārite.

$$2x=100$$

Whakawehea ngā taha e rua ki te 2.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=40$$