Whakaoti i te {l}{8a-b=9}{4a+9b=7}

$$\left. \begin{array} { l } { 8 a - b = 9 } \\ { 4 a + 9 b = 7 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{22}{19} = 1\frac{3}{19} \approx 1.157894737$$

$$b=\frac{5}{19}\approx 0.263157895$$

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.

$$8a-b=9,4a+9b=7$$

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

$$8a-b=9$$

Me tāpiri b ki ngā taha e rua o te whārite.

$$8a=b+9$$

Whakawehea ngā taha e rua ki te 8.

$$a=\frac{1}{8}\left(b+9\right)$$

Whakareatia \frac{1}{8} ki te b+9.

$$a=\frac{1}{8}b+\frac{9}{8}$$

Whakakapia te \frac{9+b}{8} mō te a ki tērā atu whārite, 4a+9b=7.

$$4\left(\frac{1}{8}b+\frac{9}{8}\right)+9b=7$$

Whakareatia 4 ki te \frac{9+b}{8}.

$$\frac{1}{2}b+\frac{9}{2}+9b=7$$

Tāpiri \frac{b}{2} ki te 9b.

$$\frac{19}{2}b+\frac{9}{2}=7$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$\frac{19}{2}b=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=\frac{5}{19}$$

Whakaurua te \frac{5}{19} mō b ki a=\frac{1}{8}b+\frac{9}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=\frac{1}{8}\times \frac{5}{19}+\frac{9}{8}$$

Whakareatia \frac{1}{8} ki te \frac{5}{19} 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.

$$a=\frac{5}{152}+\frac{9}{8}$$

Tāpiri \frac{9}{8} ki te \frac{5}{152} 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.

$$a=\frac{22}{19}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{22}{19},b=\frac{5}{19}$$

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.

$$8a-b=9,4a+9b=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-1\\4&9\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-1\\4&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-1\\4&9\end{matrix}\right))\left(\begin{matrix}8&-1\\4&9\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}8&-1\\4&9\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}8&-1\\4&9\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}8&-1\\4&9\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{9}{8\times 9-\left(-4\right)}&-\frac{-1}{8\times 9-\left(-4\right)}\\-\frac{4}{8\times 9-\left(-4\right)}&\frac{8}{8\times 9-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{9}{76}&\frac{1}{76}\\-\frac{1}{19}&\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{9}{76}\times 9+\frac{1}{76}\times 7\\-\frac{1}{19}\times 9+\frac{2}{19}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{22}{19}\\\frac{5}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{22}{19},b=\frac{5}{19}$$

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.

$$8a-b=9,4a+9b=7$$

Kia ōrite ai a 8a me 4a, 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 8.

$$4\times 8a+4\left(-1\right)b=4\times 9,8\times 4a+8\times 9b=8\times 7$$

Whakarūnātia.

$$32a-4b=36,32a+72b=56$$

Me tango 32a+72b=56 mai i 32a-4b=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$32a-32a-4b-72b=36-56$$

Tāpiri 32a ki te -32a. Ka whakakore atu ngā kupu 32a me -32a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4b-72b=36-56$$

Tāpiri -4b ki te -72b.

$$-76b=36-56$$

Tāpiri 36 ki te -56.

$$-76b=-20$$

Whakawehea ngā taha e rua ki te -76.

$$b=\frac{5}{19}$$

Whakaurua te \frac{5}{19} mō b ki 4a+9b=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$4a+9\times \frac{5}{19}=7$$

Whakareatia 9 ki te \frac{5}{19}.

$$4a+\frac{45}{19}=7$$

Me tango \frac{45}{19} mai i ngā taha e rua o te whārite.

$$4a=\frac{88}{19}$$

Whakawehea ngā taha e rua ki te 4.

$$a=\frac{22}{19}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{22}{19},b=\frac{5}{19}$$