Whakaoti i te {l}{text{(1)}8x+4y=16}{text{(2)}2(x-2y)=4}

$$\left. \begin{array} { l } { \text { (1) } 8 x + 4 y = 16 } \\ { \text { (2) } 2 ( x - 2 y ) = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 2.

$$x-2y=\frac{4}{2}$$

Whakawehea te 4 ki te 2, kia riro ko 2.

$$x-2y=2$$

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.

$$8x+4y=16,x-2y=2$$

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

$$8x+4y=16$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$8x=-4y+16$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-4y+16\right)$$

Whakareatia \frac{1}{8} ki te -4y+16.

$$x=-\frac{1}{2}y+2$$

Whakakapia te -\frac{y}{2}+2 mō te x ki tērā atu whārite, x-2y=2.

$$-\frac{1}{2}y+2-2y=2$$

Tāpiri -\frac{y}{2} ki te -2y.

$$-\frac{5}{2}y+2=2$$

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

$$-\frac{5}{2}y=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{1}{2}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 2.

$$x-2y=\frac{4}{2}$$

Whakawehea te 4 ki te 2, kia riro ko 2.

$$x-2y=2$$

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

$$8x+4y=16,x-2y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&4\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&4\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&4\\1&-2\end{matrix}\right))\left(\begin{matrix}8&4\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\1&-2\end{matrix}\right))\left(\begin{matrix}16\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\1&-2\end{matrix}\right))\left(\begin{matrix}16\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\1&-2\end{matrix}\right))\left(\begin{matrix}16\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{8\left(-2\right)-4}&-\frac{4}{8\left(-2\right)-4}\\-\frac{1}{8\left(-2\right)-4}&\frac{8}{8\left(-2\right)-4}\end{matrix}\right)\left(\begin{matrix}16\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{1}{5}\\\frac{1}{20}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}16\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 16+\frac{1}{5}\times 2\\\frac{1}{20}\times 16-\frac{2}{5}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 2.

$$x-2y=\frac{4}{2}$$

Whakawehea te 4 ki te 2, kia riro ko 2.

$$x-2y=2$$

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.

$$8x+4y=16,x-2y=2$$

Kia ōrite ai a 8x 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 8.

$$8x+4y=16,8x+8\left(-2\right)y=8\times 2$$

Whakarūnātia.

$$8x+4y=16,8x-16y=16$$

Me tango 8x-16y=16 mai i 8x+4y=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+4y+16y=16-16$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y+16y=16-16$$

Tāpiri 4y ki te 16y.

$$20y=16-16$$

Tāpiri 16 ki te -16.

$$20y=0$$

Whakawehea ngā taha e rua ki te 20.

$$y=0$$

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

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$