Whakaoti i te {l}{5b+c=8}{4b+4c=8}

$$\left. \begin{array} { l } { 5 b + c = 8 } \\ { 4 b + 4 c = 8 } \end{array} \right.$$

Whakaoti mō b, c

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

$$c=\frac{1}{2}=0.5$$

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.

$$5b+c=8,4b+4c=8$$

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

$$5b+c=8$$

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

$$5b=-c+8$$

Whakawehea ngā taha e rua ki te 5.

$$b=\frac{1}{5}\left(-c+8\right)$$

Whakareatia \frac{1}{5} ki te -c+8.

$$b=-\frac{1}{5}c+\frac{8}{5}$$

Whakakapia te \frac{-c+8}{5} mō te b ki tērā atu whārite, 4b+4c=8.

$$4\left(-\frac{1}{5}c+\frac{8}{5}\right)+4c=8$$

Whakareatia 4 ki te \frac{-c+8}{5}.

$$-\frac{4}{5}c+\frac{32}{5}+4c=8$$

Tāpiri -\frac{4c}{5} ki te 4c.

$$\frac{16}{5}c+\frac{32}{5}=8$$

Me tango \frac{32}{5} mai i ngā taha e rua o te whārite.

$$\frac{16}{5}c=\frac{8}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{16}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$c=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō c ki b=-\frac{1}{5}c+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$b=-\frac{1}{5}\times \frac{1}{2}+\frac{8}{5}$$

Whakareatia -\frac{1}{5} ki te \frac{1}{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.

$$b=-\frac{1}{10}+\frac{8}{5}$$

Tāpiri \frac{8}{5} ki te -\frac{1}{10} 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.

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

Kua oti te pūnaha te whakatau.

$$b=\frac{3}{2},c=\frac{1}{2}$$

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.

$$5b+c=8,4b+4c=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\4&4\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}8\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&1\\4&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&1\\4&4\end{matrix}\right))\left(\begin{matrix}5&1\\4&4\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\4&4\end{matrix}\right))\left(\begin{matrix}8\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\4&4\end{matrix}\right))\left(\begin{matrix}8\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\4&4\end{matrix}\right))\left(\begin{matrix}8\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-4}&-\frac{1}{5\times 4-4}\\-\frac{4}{5\times 4-4}&\frac{5}{5\times 4-4}\end{matrix}\right)\left(\begin{matrix}8\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{16}\\-\frac{1}{4}&\frac{5}{16}\end{matrix}\right)\left(\begin{matrix}8\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 8-\frac{1}{16}\times 8\\-\frac{1}{4}\times 8+\frac{5}{16}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa b me c.

$$b=\frac{3}{2},c=\frac{1}{2}$$

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.

$$5b+c=8,4b+4c=8$$

Kia ōrite ai a 5b me 4b, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$4\times 5b+4c=4\times 8,5\times 4b+5\times 4c=5\times 8$$

Whakarūnātia.

$$20b+4c=32,20b+20c=40$$

Me tango 20b+20c=40 mai i 20b+4c=32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20b-20b+4c-20c=32-40$$

Tāpiri 20b ki te -20b. Ka whakakore atu ngā kupu 20b me -20b, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4c-20c=32-40$$

Tāpiri 4c ki te -20c.

$$-16c=32-40$$

Tāpiri 32 ki te -40.

$$-16c=-8$$

Whakawehea ngā taha e rua ki te -16.

$$c=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō c ki 4b+4c=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$4b+4\times \frac{1}{2}=8$$

Whakareatia 4 ki te \frac{1}{2}.

$$4b+2=8$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$4b=6$$

Whakawehea ngā taha e rua ki te 4.

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

Kua oti te pūnaha te whakatau.

$$b=\frac{3}{2},c=\frac{1}{2}$$