Whakaoti i te {l}{y=2x}{300x+200y=7000}

$$\left. \begin{array} { l } { y = 2 x } \\ { 300 x + 200 y = 7000 } \end{array} \right.$$

Whakaoti mō y, x

$$x=10$$

$$y=20$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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.

$$y-2x=0,200y+300x=7000$$

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

$$y-2x=0$$

Me tāpiri 2x ki ngā taha e rua o te whārite.

$$y=2x$$

Whakakapia te 2x mō te y ki tērā atu whārite, 200y+300x=7000.

$$200\times 2x+300x=7000$$

Whakareatia 200 ki te 2x.

$$400x+300x=7000$$

Tāpiri 400x ki te 300x.

$$700x=7000$$

Whakawehea ngā taha e rua ki te 700.

$$x=10$$

Whakaurua te 10 mō x ki y=2x. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\times 10$$

Whakareatia 2 ki te 10.

$$y=20$$

Kua oti te pūnaha te whakatau.

$$y=20,x=10$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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

$$y-2x=0,200y+300x=7000$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\200&300\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\7000\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\200&300\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\200&300\end{matrix}\right))\left(\begin{matrix}1&-2\\200&300\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\200&300\end{matrix}\right))\left(\begin{matrix}0\\7000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\200&300\end{matrix}\right))\left(\begin{matrix}0\\7000\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\200&300\end{matrix}\right))\left(\begin{matrix}0\\7000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{300}{300-\left(-2\times 200\right)}&-\frac{-2}{300-\left(-2\times 200\right)}\\-\frac{200}{300-\left(-2\times 200\right)}&\frac{1}{300-\left(-2\times 200\right)}\end{matrix}\right)\left(\begin{matrix}0\\7000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&\frac{1}{350}\\-\frac{2}{7}&\frac{1}{700}\end{matrix}\right)\left(\begin{matrix}0\\7000\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{350}\times 7000\\\frac{1}{700}\times 7000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}20\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=20,x=10$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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.

$$y-2x=0,200y+300x=7000$$

Kia ōrite ai a y me 200y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 200 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$200y+200\left(-2\right)x=0,200y+300x=7000$$

Whakarūnātia.

$$200y-400x=0,200y+300x=7000$$

Me tango 200y+300x=7000 mai i 200y-400x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$200y-200y-400x-300x=-7000$$

Tāpiri 200y ki te -200y. Ka whakakore atu ngā kupu 200y me -200y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-400x-300x=-7000$$

Tāpiri -400x ki te -300x.

$$-700x=-7000$$

Whakawehea ngā taha e rua ki te -700.

$$x=10$$

Whakaurua te 10 mō x ki 200y+300x=7000. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$200y+300\times 10=7000$$

Whakareatia 300 ki te 10.

$$200y+3000=7000$$

Me tango 3000 mai i ngā taha e rua o te whārite.

$$200y=4000$$

Whakawehea ngā taha e rua ki te 200.

$$y=20$$

Kua oti te pūnaha te whakatau.

$$y=20,x=10$$