Whakaoti i te {l}{x-y=16}{2y=x+5}

$$\left. \begin{array} { l } { x - y = 16 } \\ { 2 y = x + 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=37$$

$$y=21$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$2y-x=5$$

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.

$$x-y=16,-x+2y=5$$

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

$$x-y=16$$

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

$$x=y+16$$

Whakakapia te y+16 mō te x ki tērā atu whārite, -x+2y=5.

$$-\left(y+16\right)+2y=5$$

Whakareatia -1 ki te y+16.

$$-y-16+2y=5$$

Tāpiri -y ki te 2y.

$$y-16=5$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$y=21$$

Whakaurua te 21 mō y ki x=y+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=21+16$$

Tāpiri 16 ki te 21.

$$x=37$$

Kua oti te pūnaha te whakatau.

$$x=37,y=21$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$2y-x=5$$

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

$$x-y=16,-x+2y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right))\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-\left(-1\right)\right)}&-\frac{-1}{2-\left(-\left(-1\right)\right)}\\-\frac{-1}{2-\left(-\left(-1\right)\right)}&\frac{1}{2-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}16\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&1\\1&1\end{matrix}\right)\left(\begin{matrix}16\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 16+5\\16+5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}37\\21\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=37,y=21$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$2y-x=5$$

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.

$$x-y=16,-x+2y=5$$

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

$$-x-\left(-y\right)=-16,-x+2y=5$$

Whakarūnātia.

$$-x+y=-16,-x+2y=5$$

Me tango -x+2y=5 mai i -x+y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x+y-2y=-16-5$$

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

$$y-2y=-16-5$$

Tāpiri y ki te -2y.

$$-y=-16-5$$

Tāpiri -16 ki te -5.

$$-y=-21$$

Whakawehea ngā taha e rua ki te -1.

$$y=21$$

Whakaurua te 21 mō y ki -x+2y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+2\times 21=5$$

Whakareatia 2 ki te 21.

$$-x+42=5$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$-x=-37$$

Whakawehea ngā taha e rua ki te -1.

$$x=37$$

Kua oti te pūnaha te whakatau.

$$x=37,y=21$$