Whakaoti i te {l}{3a+14b=4}{13a+19b=13}

$$\left\{ \begin{array} { l } { 3 a + 14 b = 4 } \\ { 13 a + 19 b = 13 } \end{array} \right.$$

Whakaoti mō a, b

$$a=\frac{106}{125}=0.848$$

$$b=\frac{13}{125}=0.104$$

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.

$$3a+14b=4,13a+19b=13$$

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

$$3a+14b=4$$

Me tango 14b mai i ngā taha e rua o te whārite.

$$3a=-14b+4$$

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{1}{3}\left(-14b+4\right)$$

Whakareatia \frac{1}{3} ki te -14b+4.

$$a=-\frac{14}{3}b+\frac{4}{3}$$

Whakakapia te \frac{-14b+4}{3} mō te a ki tērā atu whārite, 13a+19b=13.

$$13\left(-\frac{14}{3}b+\frac{4}{3}\right)+19b=13$$

Whakareatia 13 ki te \frac{-14b+4}{3}.

$$-\frac{182}{3}b+\frac{52}{3}+19b=13$$

Tāpiri -\frac{182b}{3} ki te 19b.

$$-\frac{125}{3}b+\frac{52}{3}=13$$

Me tango \frac{52}{3} mai i ngā taha e rua o te whārite.

$$-\frac{125}{3}b=-\frac{13}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{125}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=\frac{13}{125}$$

Whakaurua te \frac{13}{125} mō b ki a=-\frac{14}{3}b+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{14}{3}\times \frac{13}{125}+\frac{4}{3}$$

Whakareatia -\frac{14}{3} ki te \frac{13}{125} 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.

$$a=-\frac{182}{375}+\frac{4}{3}$$

Tāpiri \frac{4}{3} ki te -\frac{182}{375} 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.

$$a=\frac{106}{125}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{106}{125},b=\frac{13}{125}$$

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.

$$3a+14b=4,13a+19b=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&14\\13&19\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}4\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&14\\13&19\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&14\\13&19\end{matrix}\right))\left(\begin{matrix}3&14\\13&19\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&19\end{matrix}\right))\left(\begin{matrix}4\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&19\end{matrix}\right))\left(\begin{matrix}4\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&19\end{matrix}\right))\left(\begin{matrix}4\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{19}{3\times 19-14\times 13}&-\frac{14}{3\times 19-14\times 13}\\-\frac{13}{3\times 19-14\times 13}&\frac{3}{3\times 19-14\times 13}\end{matrix}\right)\left(\begin{matrix}4\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{19}{125}&\frac{14}{125}\\\frac{13}{125}&-\frac{3}{125}\end{matrix}\right)\left(\begin{matrix}4\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{19}{125}\times 4+\frac{14}{125}\times 13\\\frac{13}{125}\times 4-\frac{3}{125}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{106}{125}\\\frac{13}{125}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{106}{125},b=\frac{13}{125}$$

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.

$$3a+14b=4,13a+19b=13$$

Kia ōrite ai a 3a me 13a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 13 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$13\times 3a+13\times 14b=13\times 4,3\times 13a+3\times 19b=3\times 13$$

Whakarūnātia.

$$39a+182b=52,39a+57b=39$$

Me tango 39a+57b=39 mai i 39a+182b=52 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$39a-39a+182b-57b=52-39$$

Tāpiri 39a ki te -39a. Ka whakakore atu ngā kupu 39a me -39a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$182b-57b=52-39$$

Tāpiri 182b ki te -57b.

$$125b=52-39$$

Tāpiri 52 ki te -39.

$$125b=13$$

Whakawehea ngā taha e rua ki te 125.

$$b=\frac{13}{125}$$

Whakaurua te \frac{13}{125} mō b ki 13a+19b=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$13a+19\times \frac{13}{125}=13$$

Whakareatia 19 ki te \frac{13}{125}.

$$13a+\frac{247}{125}=13$$

Me tango \frac{247}{125} mai i ngā taha e rua o te whārite.

$$13a=\frac{1378}{125}$$

Whakawehea ngā taha e rua ki te 13.

$$a=\frac{106}{125}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{106}{125},b=\frac{13}{125}$$