Whakaoti i te {l}{30f_{1}+40f_{2}=285}{30f_{1}+30f_{2}=+270}

$$\left. \begin{array} { l } { 30 f _ { 1 } + 40 f _ { 2 } = 285 } \\ { 30 f _ { 1 } + 30 f _ { 2 } = + 270 } \end{array} \right.$$

Whakaoti mō f_1, f_2

$$f_{1} = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

$$f_{2} = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

$$30f_{1}+40f_{2}=285,30f_{1}+30f_{2}=270$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te f_{1} mā te wehe i te f_{1} i te taha mauī o te tohu ōrite.

$$30f_{1}+40f_{2}=285$$

Me tango 40f_{2} mai i ngā taha e rua o te whārite.

$$30f_{1}=-40f_{2}+285$$

Whakawehea ngā taha e rua ki te 30.

$$f_{1}=\frac{1}{30}\left(-40f_{2}+285\right)$$

Whakareatia \frac{1}{30} ki te -40f_{2}+285.

$$f_{1}=-\frac{4}{3}f_{2}+\frac{19}{2}$$

Whakakapia te -\frac{4f_{2}}{3}+\frac{19}{2} mō te f_{1} ki tērā atu whārite, 30f_{1}+30f_{2}=270.

$$30\left(-\frac{4}{3}f_{2}+\frac{19}{2}\right)+30f_{2}=270$$

Whakareatia 30 ki te -\frac{4f_{2}}{3}+\frac{19}{2}.

$$-40f_{2}+285+30f_{2}=270$$

Tāpiri -40f_{2} ki te 30f_{2}.

$$-10f_{2}+285=270$$

Me tango 285 mai i ngā taha e rua o te whārite.

$$-10f_{2}=-15$$

Whakawehea ngā taha e rua ki te -10.

$$f_{2}=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō f_{2} ki f_{1}=-\frac{4}{3}f_{2}+\frac{19}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō f_{1} hāngai tonu.

$$f_{1}=-\frac{4}{3}\times \frac{3}{2}+\frac{19}{2}$$

Whakareatia -\frac{4}{3} ki te \frac{3}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$f_{1}=-2+\frac{19}{2}$$

Tāpiri \frac{19}{2} ki te -2.

$$f_{1}=\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$f_{1}=\frac{15}{2},f_{2}=\frac{3}{2}$$

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.

$$30f_{1}+40f_{2}=285,30f_{1}+30f_{2}=270$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}30&40\\30&30\end{matrix}\right)\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=\left(\begin{matrix}285\\270\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}30&40\\30&30\end{matrix}\right).

$$inverse(\left(\begin{matrix}30&40\\30&30\end{matrix}\right))\left(\begin{matrix}30&40\\30&30\end{matrix}\right)\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}30&40\\30&30\end{matrix}\right))\left(\begin{matrix}285\\270\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}30&40\\30&30\end{matrix}\right))\left(\begin{matrix}285\\270\end{matrix}\right)$$

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

$$\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}30&40\\30&30\end{matrix}\right))\left(\begin{matrix}285\\270\end{matrix}\right)$$

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.

$$\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{30}{30\times 30-40\times 30}&-\frac{40}{30\times 30-40\times 30}\\-\frac{30}{30\times 30-40\times 30}&\frac{30}{30\times 30-40\times 30}\end{matrix}\right)\left(\begin{matrix}285\\270\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}&\frac{2}{15}\\\frac{1}{10}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}285\\270\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}\times 285+\frac{2}{15}\times 270\\\frac{1}{10}\times 285-\frac{1}{10}\times 270\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}f_{1}\\f_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{15}{2}\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa f_{1} me f_{2}.

$$f_{1}=\frac{15}{2},f_{2}=\frac{3}{2}$$

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.

$$30f_{1}+40f_{2}=285,30f_{1}+30f_{2}=270$$

Me tango 30f_{1}+30f_{2}=270 mai i 30f_{1}+40f_{2}=285 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30f_{1}-30f_{1}+40f_{2}-30f_{2}=285-270$$

Tāpiri 30f_{1} ki te -30f_{1}. Ka whakakore atu ngā kupu 30f_{1} me -30f_{1}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40f_{2}-30f_{2}=285-270$$

Tāpiri 40f_{2} ki te -30f_{2}.

$$10f_{2}=285-270$$

Tāpiri 285 ki te -270.

$$10f_{2}=15$$

Whakawehea ngā taha e rua ki te 10.

$$f_{2}=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō f_{2} ki 30f_{1}+30f_{2}=270. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō f_{1} hāngai tonu.

$$30f_{1}+30\times \frac{3}{2}=270$$

Whakareatia 30 ki te \frac{3}{2}.

$$30f_{1}+45=270$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$30f_{1}=225$$

Whakawehea ngā taha e rua ki te 30.

$$f_{1}=\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$f_{1}=\frac{15}{2},f_{2}=\frac{3}{2}$$