Whakaoti i te {l}{-9a+7b=2}{-5=3b+5a}

$$\left. \begin{array} { l } { - 9 a + 7 b = 2 } \\ { - 5 = 3 b + 5 a } \end{array} \right.$$

Whakaoti mō a, b

$$a=-\frac{41}{62}\approx -0.661290323$$

$$b=-\frac{35}{62}\approx -0.564516129$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3b+5a=-5$$

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.

$$-9a+7b=2,5a+3b=-5$$

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

$$-9a+7b=2$$

Me tango 7b mai i ngā taha e rua o te whārite.

$$-9a=-7b+2$$

Whakawehea ngā taha e rua ki te -9.

$$a=-\frac{1}{9}\left(-7b+2\right)$$

Whakareatia -\frac{1}{9} ki te -7b+2.

$$a=\frac{7}{9}b-\frac{2}{9}$$

Whakakapia te \frac{7b-2}{9} mō te a ki tērā atu whārite, 5a+3b=-5.

$$5\left(\frac{7}{9}b-\frac{2}{9}\right)+3b=-5$$

Whakareatia 5 ki te \frac{7b-2}{9}.

$$\frac{35}{9}b-\frac{10}{9}+3b=-5$$

Tāpiri \frac{35b}{9} ki te 3b.

$$\frac{62}{9}b-\frac{10}{9}=-5$$

Me tāpiri \frac{10}{9} ki ngā taha e rua o te whārite.

$$\frac{62}{9}b=-\frac{35}{9}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{62}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-\frac{35}{62}$$

Whakaurua te -\frac{35}{62} mō b ki a=\frac{7}{9}b-\frac{2}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=\frac{7}{9}\left(-\frac{35}{62}\right)-\frac{2}{9}$$

Whakareatia \frac{7}{9} ki te -\frac{35}{62} 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.

$$a=-\frac{245}{558}-\frac{2}{9}$$

Tāpiri -\frac{2}{9} ki te -\frac{245}{558} 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.

$$a=-\frac{41}{62}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{41}{62},b=-\frac{35}{62}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3b+5a=-5$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-9a+7b=2,5a+3b=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&7\\5&3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}2\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&7\\5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&7\\5&3\end{matrix}\right))\left(\begin{matrix}-9&7\\5&3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-9&7\\5&3\end{matrix}\right))\left(\begin{matrix}2\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-9&7\\5&3\end{matrix}\right))\left(\begin{matrix}2\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-9&7\\5&3\end{matrix}\right))\left(\begin{matrix}2\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-9\times 3-7\times 5}&-\frac{7}{-9\times 3-7\times 5}\\-\frac{5}{-9\times 3-7\times 5}&-\frac{9}{-9\times 3-7\times 5}\end{matrix}\right)\left(\begin{matrix}2\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{62}&\frac{7}{62}\\\frac{5}{62}&\frac{9}{62}\end{matrix}\right)\left(\begin{matrix}2\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{62}\times 2+\frac{7}{62}\left(-5\right)\\\frac{5}{62}\times 2+\frac{9}{62}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{41}{62}\\-\frac{35}{62}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-\frac{41}{62},b=-\frac{35}{62}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3b+5a=-5$$

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.

$$-9a+7b=2,5a+3b=-5$$

Kia ōrite ai a -9a me 5a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -9.

$$5\left(-9\right)a+5\times 7b=5\times 2,-9\times 5a-9\times 3b=-9\left(-5\right)$$

Whakarūnātia.

$$-45a+35b=10,-45a-27b=45$$

Me tango -45a-27b=45 mai i -45a+35b=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-45a+45a+35b+27b=10-45$$

Tāpiri -45a ki te 45a. Ka whakakore atu ngā kupu -45a me 45a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$35b+27b=10-45$$

Tāpiri 35b ki te 27b.

$$62b=10-45$$

Tāpiri 10 ki te -45.

$$62b=-35$$

Whakawehea ngā taha e rua ki te 62.

$$b=-\frac{35}{62}$$

Whakaurua te -\frac{35}{62} mō b ki 5a+3b=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$5a+3\left(-\frac{35}{62}\right)=-5$$

Whakareatia 3 ki te -\frac{35}{62}.

$$5a-\frac{105}{62}=-5$$

Me tāpiri \frac{105}{62} ki ngā taha e rua o te whārite.

$$5a=-\frac{205}{62}$$

Whakawehea ngā taha e rua ki te 5.

$$a=-\frac{41}{62}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{41}{62},b=-\frac{35}{62}$$