Whakaoti i te {l}{x+2y=5900}{2x+2y=9400}

$$\left\{ \begin{array} { l } { x + 2 y = 5900 } \\ { 2 x + 2 y = 9400 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3500$$

$$y=1200$$

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.

$$x+2y=5900,2x+2y=9400$$

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

$$x+2y=5900$$

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

$$x=-2y+5900$$

Whakakapia te -2y+5900 mō te x ki tērā atu whārite, 2x+2y=9400.

$$2\left(-2y+5900\right)+2y=9400$$

Whakareatia 2 ki te -2y+5900.

$$-4y+11800+2y=9400$$

Tāpiri -4y ki te 2y.

$$-2y+11800=9400$$

Me tango 11800 mai i ngā taha e rua o te whārite.

$$-2y=-2400$$

Whakawehea ngā taha e rua ki te -2.

$$y=1200$$

Whakaurua te 1200 mō y ki x=-2y+5900. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 1200+5900$$

Whakareatia -2 ki te 1200.

$$x=-2400+5900$$

Tāpiri 5900 ki te -2400.

$$x=3500$$

Kua oti te pūnaha te whakatau.

$$x=3500,y=1200$$

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.

$$x+2y=5900,2x+2y=9400$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5900\\9400\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\2&2\end{matrix}\right))\left(\begin{matrix}1&2\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&2\end{matrix}\right))\left(\begin{matrix}5900\\9400\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&2\end{matrix}\right))\left(\begin{matrix}5900\\9400\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&2\end{matrix}\right))\left(\begin{matrix}5900\\9400\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-2\times 2}&-\frac{2}{2-2\times 2}\\-\frac{2}{2-2\times 2}&\frac{1}{2-2\times 2}\end{matrix}\right)\left(\begin{matrix}5900\\9400\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&1\\1&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}5900\\9400\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5900+9400\\5900-\frac{1}{2}\times 9400\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3500\\1200\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3500,y=1200$$

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.

$$x+2y=5900,2x+2y=9400$$

Me tango 2x+2y=9400 mai i x+2y=5900 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-2x+2y-2y=5900-9400$$

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.

$$x-2x=5900-9400$$

Tāpiri x ki te -2x.

$$-x=5900-9400$$

Tāpiri 5900 ki te -9400.

$$-x=-3500$$

Whakawehea ngā taha e rua ki te -1.

$$x=3500$$

Whakaurua te 3500 mō x ki 2x+2y=9400. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\times 3500+2y=9400$$

Whakareatia 2 ki te 3500.

$$7000+2y=9400$$

Me tango 7000 mai i ngā taha e rua o te whārite.

$$2y=2400$$

Whakawehea ngā taha e rua ki te 2.

$$y=1200$$

Kua oti te pūnaha te whakatau.

$$x=3500,y=1200$$