Whakaoti i te {l}{x+y=8}{2y=x+1}

$$\left. \begin{array} { l } { x + y = 8 } \\ { 2 y = x + 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$2y-x=1$$

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.

$$x+y=8,-x+2y=1$$

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

$$x+y=8$$

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

$$x=-y+8$$

Whakakapia te -y+8 mō te x ki tērā atu whārite, -x+2y=1.

$$-\left(-y+8\right)+2y=1$$

Whakareatia -1 ki te -y+8.

$$y-8+2y=1$$

Tāpiri y ki te 2y.

$$3y-8=1$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$3y=9$$

Whakawehea ngā taha e rua ki te 3.

$$y=3$$

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

$$x=-3+8$$

Tāpiri 8 ki te -3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$2y-x=1$$

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

$$x+y=8,-x+2y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\-1&2\end{matrix}\right))\left(\begin{matrix}1&1\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&2\end{matrix}\right))\left(\begin{matrix}8\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&2\end{matrix}\right))\left(\begin{matrix}8\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&2\end{matrix}\right))\left(\begin{matrix}8\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-1\right)}&-\frac{1}{2-\left(-1\right)}\\-\frac{-1}{2-\left(-1\right)}&\frac{1}{2-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}8\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&-\frac{1}{3}\\\frac{1}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}8\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 8-\frac{1}{3}\\\frac{1}{3}\times 8+\frac{1}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$2y-x=1$$

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.

$$x+y=8,-x+2y=1$$

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.

$$-x-y=-8,-x+2y=1$$

Me tango -x+2y=1 mai i -x-y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x-y-2y=-8-1$$

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.

$$-y-2y=-8-1$$

Tāpiri -y ki te -2y.

$$-3y=-8-1$$

Tāpiri -8 ki te -1.

$$-3y=-9$$

Whakawehea ngā taha e rua ki te -3.

$$y=3$$

Whakaurua te 3 mō y ki -x+2y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+2\times 3=1$$

Whakareatia 2 ki te 3.

$$-x+6=1$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-x=-5$$

Whakawehea ngā taha e rua ki te -1.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=3$$