Whakaoti i te {l}{x-3y=1}{2x+5y=90}

$$\left. \begin{array} { l } { x - 3 y = 1 } \\ { 2 x + 5 y = 90 } \end{array} \right.$$

Whakaoti mō x, y

$$x=25$$

$$y=8$$

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.

$$x-3y=1,2x+5y=90$$

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

$$x-3y=1$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y+1$$

Whakakapia te 3y+1 mō te x ki tērā atu whārite, 2x+5y=90.

$$2\left(3y+1\right)+5y=90$$

Whakareatia 2 ki te 3y+1.

$$6y+2+5y=90$$

Tāpiri 6y ki te 5y.

$$11y+2=90$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$11y=88$$

Whakawehea ngā taha e rua ki te 11.

$$y=8$$

Whakaurua te 8 mō y ki x=3y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times 8+1$$

Whakareatia 3 ki te 8.

$$x=24+1$$

Tāpiri 1 ki te 24.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=8$$

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.

$$x-3y=1,2x+5y=90$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\90\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\2&5\end{matrix}\right))\left(\begin{matrix}1&-3\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\2&5\end{matrix}\right))\left(\begin{matrix}1\\90\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\2&5\end{matrix}\right))\left(\begin{matrix}1\\90\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\2&5\end{matrix}\right))\left(\begin{matrix}1\\90\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-3\times 2\right)}&-\frac{-3}{5-\left(-3\times 2\right)}\\-\frac{2}{5-\left(-3\times 2\right)}&\frac{1}{5-\left(-3\times 2\right)}\end{matrix}\right)\left(\begin{matrix}1\\90\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{11}&\frac{3}{11}\\-\frac{2}{11}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}1\\90\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{11}+\frac{3}{11}\times 90\\-\frac{2}{11}+\frac{1}{11}\times 90\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=25,y=8$$

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.

$$x-3y=1,2x+5y=90$$

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.

$$2x+2\left(-3\right)y=2,2x+5y=90$$

Whakarūnātia.

$$2x-6y=2,2x+5y=90$$

Me tango 2x+5y=90 mai i 2x-6y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-6y-5y=2-90$$

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.

$$-6y-5y=2-90$$

Tāpiri -6y ki te -5y.

$$-11y=2-90$$

Tāpiri 2 ki te -90.

$$-11y=-88$$

Whakawehea ngā taha e rua ki te -11.

$$y=8$$

Whakaurua te 8 mō y ki 2x+5y=90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\times 8=90$$

Whakareatia 5 ki te 8.

$$2x+40=90$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$2x=50$$

Whakawehea ngā taha e rua ki te 2.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=8$$