Whakaoti i te {l}{a+3b=30}{3a+5b=30}

$$\left\{ \begin{array} { l } { a + 3 b = 30 } \\ { 3 a + 5 b = 30 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-15$$

$$b=15$$

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.

$$a+3b=30,3a+5b=30$$

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

$$a+3b=30$$

Me tango 3b mai i ngā taha e rua o te whārite.

$$a=-3b+30$$

Whakakapia te -3b+30 mō te a ki tērā atu whārite, 3a+5b=30.

$$3\left(-3b+30\right)+5b=30$$

Whakareatia 3 ki te -3b+30.

$$-9b+90+5b=30$$

Tāpiri -9b ki te 5b.

$$-4b+90=30$$

Me tango 90 mai i ngā taha e rua o te whārite.

$$-4b=-60$$

Whakawehea ngā taha e rua ki te -4.

$$b=15$$

Whakaurua te 15 mō b ki a=-3b+30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-3\times 15+30$$

Whakareatia -3 ki te 15.

$$a=-45+30$$

Tāpiri 30 ki te -45.

$$a=-15$$

Kua oti te pūnaha te whakatau.

$$a=-15,b=15$$

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.

$$a+3b=30,3a+5b=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\3&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}30\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&3\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&3\\3&5\end{matrix}\right))\left(\begin{matrix}1&3\\3&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\3&5\end{matrix}\right))\left(\begin{matrix}30\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\3&5\end{matrix}\right))\left(\begin{matrix}30\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\3&5\end{matrix}\right))\left(\begin{matrix}30\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-3\times 3}&-\frac{3}{5-3\times 3}\\-\frac{3}{5-3\times 3}&\frac{1}{5-3\times 3}\end{matrix}\right)\left(\begin{matrix}30\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4}&\frac{3}{4}\\\frac{3}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}30\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4}\times 30+\frac{3}{4}\times 30\\\frac{3}{4}\times 30-\frac{1}{4}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-15\\15\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-15,b=15$$

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.

$$a+3b=30,3a+5b=30$$

Kia ōrite ai a a 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 1.

$$3a+3\times 3b=3\times 30,3a+5b=30$$

Whakarūnātia.

$$3a+9b=90,3a+5b=30$$

Me tango 3a+5b=30 mai i 3a+9b=90 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3a-3a+9b-5b=90-30$$

Tāpiri 3a ki te -3a. Ka whakakore atu ngā kupu 3a me -3a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9b-5b=90-30$$

Tāpiri 9b ki te -5b.

$$4b=90-30$$

Tāpiri 90 ki te -30.

$$4b=60$$

Whakawehea ngā taha e rua ki te 4.

$$b=15$$

Whakaurua te 15 mō b ki 3a+5b=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$3a+5\times 15=30$$

Whakareatia 5 ki te 15.

$$3a+75=30$$

Me tango 75 mai i ngā taha e rua o te whārite.

$$3a=-45$$

Whakawehea ngā taha e rua ki te 3.

$$a=-15$$

Kua oti te pūnaha te whakatau.

$$a=-15,b=15$$