Whakaoti i te {r}{d+q=40}{10d+0.25q=5.80}

$$\left. \begin{array} { r } { d + q = 40 } \\ { 10 d + 0.25 q = 5.80 } \end{array} \right.$$

Whakaoti mō d, q

$$d=-\frac{28}{65}\approx -0.430769231$$

$$q = \frac{2628}{65} = 40\frac{28}{65} \approx 40.430769231$$

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.

$$d+q=40,10d+0.25q=5.8$$

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

$$d+q=40$$

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

$$d=-q+40$$

Whakakapia te -q+40 mō te d ki tērā atu whārite, 10d+0.25q=5.8.

$$10\left(-q+40\right)+0.25q=5.8$$

Whakareatia 10 ki te -q+40.

$$-10q+400+0.25q=5.8$$

Tāpiri -10q ki te \frac{q}{4}.

$$-9.75q+400=5.8$$

Me tango 400 mai i ngā taha e rua o te whārite.

$$-9.75q=-394.2$$

Whakawehea ngā taha e rua o te whārite ki te -9.75, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$q=\frac{2628}{65}$$

Whakaurua te \frac{2628}{65} mō q ki d=-q+40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō d hāngai tonu.

$$d=-\frac{2628}{65}+40$$

Tāpiri 40 ki te -\frac{2628}{65}.

$$d=-\frac{28}{65}$$

Kua oti te pūnaha te whakatau.

$$d=-\frac{28}{65},q=\frac{2628}{65}$$

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.

$$d+q=40,10d+0.25q=5.8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\10&0.25\end{matrix}\right)\left(\begin{matrix}d\\q\end{matrix}\right)=\left(\begin{matrix}40\\5.8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\10&0.25\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\10&0.25\end{matrix}\right))\left(\begin{matrix}1&1\\10&0.25\end{matrix}\right)\left(\begin{matrix}d\\q\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\10&0.25\end{matrix}\right))\left(\begin{matrix}40\\5.8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}d\\q\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\10&0.25\end{matrix}\right))\left(\begin{matrix}40\\5.8\end{matrix}\right)$$

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

$$\left(\begin{matrix}d\\q\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\10&0.25\end{matrix}\right))\left(\begin{matrix}40\\5.8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}d\\q\end{matrix}\right)=\left(\begin{matrix}\frac{0.25}{0.25-10}&-\frac{1}{0.25-10}\\-\frac{10}{0.25-10}&\frac{1}{0.25-10}\end{matrix}\right)\left(\begin{matrix}40\\5.8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}d\\q\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{39}&\frac{4}{39}\\\frac{40}{39}&-\frac{4}{39}\end{matrix}\right)\left(\begin{matrix}40\\5.8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}d\\q\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{39}\times 40+\frac{4}{39}\times 5.8\\\frac{40}{39}\times 40-\frac{4}{39}\times 5.8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}d\\q\end{matrix}\right)=\left(\begin{matrix}-\frac{28}{65}\\\frac{2628}{65}\end{matrix}\right)$$

Tangohia ngā huānga poukapa d me q.

$$d=-\frac{28}{65},q=\frac{2628}{65}$$

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.

$$d+q=40,10d+0.25q=5.8$$

Kia ōrite ai a d me 10d, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$10d+10q=10\times 40,10d+0.25q=5.8$$

Whakarūnātia.

$$10d+10q=400,10d+0.25q=5.8$$

Me tango 10d+0.25q=5.8 mai i 10d+10q=400 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10d-10d+10q-0.25q=400-5.8$$

Tāpiri 10d ki te -10d. Ka whakakore atu ngā kupu 10d me -10d, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10q-0.25q=400-5.8$$

Tāpiri 10q ki te -\frac{q}{4}.

$$9.75q=400-5.8$$

Tāpiri 400 ki te -5.8.

$$9.75q=394.2$$

Whakawehea ngā taha e rua o te whārite ki te 9.75, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$q=\frac{2628}{65}$$

Whakaurua te \frac{2628}{65} mō q ki 10d+0.25q=5.8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō d hāngai tonu.

$$10d+0.25\times \frac{2628}{65}=5.8$$

Whakareatia 0.25 ki te \frac{2628}{65} 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.

$$10d+\frac{657}{65}=5.8$$

Me tango \frac{657}{65} mai i ngā taha e rua o te whārite.

$$10d=-\frac{56}{13}$$

Whakawehea ngā taha e rua ki te 10.

$$d=-\frac{28}{65}$$

Kua oti te pūnaha te whakatau.

$$d=-\frac{28}{65},q=\frac{2628}{65}$$