Whakaoti i te {l}{3.5c+5.25a=365.75}{c+a=9.4}

$$\left. \begin{array} { l } { 3.5 c + 5.25 a = 365.75 } \\ { c + a = 9.4 } \end{array} \right.$$

Whakaoti mō c, a

$$c=-180.8$$

$$a=190.2$$

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.

$$3.5c+5.25a=365.75,c+a=9.4$$

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

$$3.5c+5.25a=365.75$$

Me tango \frac{21a}{4} mai i ngā taha e rua o te whārite.

$$3.5c=-5.25a+365.75$$

Whakawehea ngā taha e rua o te whārite ki te 3.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$c=\frac{2}{7}\left(-5.25a+365.75\right)$$

Whakareatia \frac{2}{7} ki te \frac{-21a+1463}{4}.

$$c=-1.5a+104.5$$

Whakakapia te \frac{-3a+209}{2} mō te c ki tērā atu whārite, c+a=9.4.

$$-1.5a+104.5+a=9.4$$

Tāpiri -\frac{3a}{2} ki te a.

$$-0.5a+104.5=9.4$$

Me tango 104.5 mai i ngā taha e rua o te whārite.

$$-0.5a=-95.1$$

Me whakarea ngā taha e rua ki te -2.

$$a=190.2$$

Whakaurua te 190.2 mō a ki c=-1.5a+104.5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c=-1.5\times 190.2+104.5$$

Whakareatia -1.5 ki te 190.2 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.

$$c=-285.3+104.5$$

Tāpiri 104.5 ki te -285.3 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$c=-180.8$$

Kua oti te pūnaha te whakatau.

$$c=-180.8,a=190.2$$

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.

$$3.5c+5.25a=365.75,c+a=9.4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right)\left(\begin{matrix}c\\a\end{matrix}\right)=\left(\begin{matrix}365.75\\9.4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right))\left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right)\left(\begin{matrix}c\\a\end{matrix}\right)=inverse(\left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right))\left(\begin{matrix}365.75\\9.4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\a\end{matrix}\right)=inverse(\left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right))\left(\begin{matrix}365.75\\9.4\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\a\end{matrix}\right)=inverse(\left(\begin{matrix}3.5&5.25\\1&1\end{matrix}\right))\left(\begin{matrix}365.75\\9.4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\a\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3.5-5.25}&-\frac{5.25}{3.5-5.25}\\-\frac{1}{3.5-5.25}&\frac{3.5}{3.5-5.25}\end{matrix}\right)\left(\begin{matrix}365.75\\9.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}&3\\\frac{4}{7}&-2\end{matrix}\right)\left(\begin{matrix}365.75\\9.4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}c\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}\times 365.75+3\times 9.4\\\frac{4}{7}\times 365.75-2\times 9.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\a\end{matrix}\right)=\left(\begin{matrix}-180.8\\190.2\end{matrix}\right)$$

Tangohia ngā huānga poukapa c me a.

$$c=-180.8,a=190.2$$

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.

$$3.5c+5.25a=365.75,c+a=9.4$$

Kia ōrite ai a \frac{7c}{2} me c, 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 3.5.

$$3.5c+5.25a=365.75,3.5c+3.5a=3.5\times 9.4$$

Whakarūnātia.

$$3.5c+5.25a=365.75,3.5c+3.5a=32.9$$

Me tango 3.5c+3.5a=32.9 mai i 3.5c+5.25a=365.75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3.5c-3.5c+5.25a-3.5a=365.75-32.9$$

Tāpiri \frac{7c}{2} ki te -\frac{7c}{2}. Ka whakakore atu ngā kupu \frac{7c}{2} me -\frac{7c}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5.25a-3.5a=365.75-32.9$$

Tāpiri \frac{21a}{4} ki te -\frac{7a}{2}.

$$1.75a=365.75-32.9$$

Tāpiri 365.75 ki te -32.9 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$1.75a=332.85$$

Whakawehea ngā taha e rua o te whārite ki te 1.75, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$a=190.2$$

Whakaurua te 190.2 mō a ki c+a=9.4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c+190.2=9.4$$

Me tango 190.2 mai i ngā taha e rua o te whārite.

$$c=-180.8$$

Kua oti te pūnaha te whakatau.

$$c=-180.8,a=190.2$$