Whakaoti i te {l}{7a-10b=-64}{5b+3a=19}

$$\left\{ \begin{array} { l } { 7 a - 10 b = - 64 } \\ { 5 b + 3 a = 19 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-2$$

$$b=5$$

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.

$$7a-10b=-64,3a+5b=19$$

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

$$7a-10b=-64$$

Me tāpiri 10b ki ngā taha e rua o te whārite.

$$7a=10b-64$$

Whakawehea ngā taha e rua ki te 7.

$$a=\frac{1}{7}\left(10b-64\right)$$

Whakareatia \frac{1}{7} ki te 10b-64.

$$a=\frac{10}{7}b-\frac{64}{7}$$

Whakakapia te \frac{10b-64}{7} mō te a ki tērā atu whārite, 3a+5b=19.

$$3\left(\frac{10}{7}b-\frac{64}{7}\right)+5b=19$$

Whakareatia 3 ki te \frac{10b-64}{7}.

$$\frac{30}{7}b-\frac{192}{7}+5b=19$$

Tāpiri \frac{30b}{7} ki te 5b.

$$\frac{65}{7}b-\frac{192}{7}=19$$

Me tāpiri \frac{192}{7} ki ngā taha e rua o te whārite.

$$\frac{65}{7}b=\frac{325}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{65}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=5$$

Whakaurua te 5 mō b ki a=\frac{10}{7}b-\frac{64}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=\frac{10}{7}\times 5-\frac{64}{7}$$

Whakareatia \frac{10}{7} ki te 5.

$$a=\frac{50-64}{7}$$

Tāpiri -\frac{64}{7} ki te \frac{50}{7} 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.

$$a=-2$$

Kua oti te pūnaha te whakatau.

$$a=-2,b=5$$

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.

$$7a-10b=-64,3a+5b=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-10\\3&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-10\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}7&-10\\3&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}-64\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}-64\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7\times 5-\left(-10\times 3\right)}&-\frac{-10}{7\times 5-\left(-10\times 3\right)}\\-\frac{3}{7\times 5-\left(-10\times 3\right)}&\frac{7}{7\times 5-\left(-10\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{2}{13}\\-\frac{3}{65}&\frac{7}{65}\end{matrix}\right)\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\left(-64\right)+\frac{2}{13}\times 19\\-\frac{3}{65}\left(-64\right)+\frac{7}{65}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=-2,b=5$$

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.

$$7a-10b=-64,3a+5b=19$$

Kia ōrite ai a 7a me 3a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$3\times 7a+3\left(-10\right)b=3\left(-64\right),7\times 3a+7\times 5b=7\times 19$$

Whakarūnātia.

$$21a-30b=-192,21a+35b=133$$

Me tango 21a+35b=133 mai i 21a-30b=-192 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$21a-21a-30b-35b=-192-133$$

Tāpiri 21a ki te -21a. Ka whakakore atu ngā kupu 21a me -21a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-30b-35b=-192-133$$

Tāpiri -30b ki te -35b.

$$-65b=-192-133$$

Tāpiri -192 ki te -133.

$$-65b=-325$$

Whakawehea ngā taha e rua ki te -65.

$$b=5$$

Whakaurua te 5 mō b ki 3a+5b=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$3a+5\times 5=19$$

Whakareatia 5 ki te 5.

$$3a+25=19$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$3a=-6$$

Whakawehea ngā taha e rua ki te 3.

$$a=-2$$

Kua oti te pūnaha te whakatau.

$$a=-2,b=5$$