Whakaoti i te f/11=g/33,text{siendo}f+g=40

$$\frac { f } { 11 } = \frac { g } { 33 } , \text { siendo } f + g = 40$$

Whakaoti mō f, g

$$f=10$$

$$g=30$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 33, arā, te tauraro pātahi he tino iti rawa te kitea o 11,33.

$$3f=g$$

Whakawehea ngā taha e rua ki te 3.

$$f=\frac{1}{3}g$$

Whakakapia te \frac{g}{3} mō te f ki tērā atu whārite, f+g=40.

$$\frac{1}{3}g+g=40$$

Tāpiri \frac{g}{3} ki te g.

$$\frac{4}{3}g=40$$

Whakawehea ngā taha e rua o te whārite ki te \frac{4}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$g=30$$

Whakaurua te 30 mō g ki f=\frac{1}{3}g. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō f hāngai tonu.

$$f=\frac{1}{3}\times 30$$

Whakareatia \frac{1}{3} ki te 30.

$$f=10$$

Kua oti te pūnaha te whakatau.

$$f=10,g=30$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 33, arā, te tauraro pātahi he tino iti rawa te kitea o 11,33.

$$3f=g$$

Tangohia te g mai i ngā taha e rua.

$$3f-g=0$$

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

$$3f-g=0,f+g=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\1&1\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}0\\40\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}3&-1\\1&1\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}0\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}0\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}f\\g\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}0\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-1\right)}&-\frac{-1}{3-\left(-1\right)}\\-\frac{1}{3-\left(-1\right)}&\frac{3}{3-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}0\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{4}&\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}0\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 40\\\frac{3}{4}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}10\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa f me g.

$$f=10,g=30$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 33, arā, te tauraro pātahi he tino iti rawa te kitea o 11,33.

$$3f=g$$

Tangohia te g mai i ngā taha e rua.

$$3f-g=0$$

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.

$$3f-g=0,f+g=40$$

Kia ōrite ai a 3f me f, 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 3.

$$3f-g=0,3f+3g=3\times 40$$

Whakarūnātia.

$$3f-g=0,3f+3g=120$$

Me tango 3f+3g=120 mai i 3f-g=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3f-3f-g-3g=-120$$

Tāpiri 3f ki te -3f. Ka whakakore atu ngā kupu 3f me -3f, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-g-3g=-120$$

Tāpiri -g ki te -3g.

$$-4g=-120$$

Whakawehea ngā taha e rua ki te -4.

$$g=30$$

Whakaurua te 30 mō g ki f+g=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō f hāngai tonu.

$$f+30=40$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$f=10$$

Kua oti te pūnaha te whakatau.

$$f=10,g=30$$