Whakaoti i te {c}{y=3+x}{5x-2y=0}

$$\left. \begin{array} { c } { y = 3 + x } \\ { 5 x - 2 y = 0 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=3$$

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.

$$y-x=3,-2y+5x=0$$

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

$$y-x=3$$

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

$$y=x+3$$

Whakakapia te x+3 mō te y ki tērā atu whārite, -2y+5x=0.

$$-2\left(x+3\right)+5x=0$$

Whakareatia -2 ki te x+3.

$$-2x-6+5x=0$$

Tāpiri -2x ki te 5x.

$$3x-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

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

$$y=2+3$$

Tāpiri 3 ki te 2.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=3$$

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

$$y-x=3,-2y+5x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=5,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=3$$

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.

$$y-x=3,-2y+5x=0$$

Kia ōrite ai a y me -2y, 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.

$$-2y-2\left(-1\right)x=-2\times 3,-2y+5x=0$$

Whakarūnātia.

$$-2y+2x=-6,-2y+5x=0$$

Me tango -2y+5x=0 mai i -2y+2x=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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

$$2x-5x=-6$$

Tāpiri 2x ki te -5x.

$$-3x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$x=2$$

Whakaurua te 2 mō x ki -2y+5x=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2y+5\times 2=0$$

Whakareatia 5 ki te 2.

$$-2y+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-2y=-10$$

Whakawehea ngā taha e rua ki te -2.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$