Whakaoti i te {l}{a+2b=29}{2a+b=17}

$$\left. \begin{array} { l } { a + 2 b = 29 } \\ { 2 a + b = 17 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$b = \frac{41}{3} = 13\frac{2}{3} \approx 13.666666667$$

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.

$$a+2b=29,2a+b=17$$

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

$$a+2b=29$$

Me tango 2b mai i ngā taha e rua o te whārite.

$$a=-2b+29$$

Whakakapia te -2b+29 mō te a ki tērā atu whārite, 2a+b=17.

$$2\left(-2b+29\right)+b=17$$

Whakareatia 2 ki te -2b+29.

$$-4b+58+b=17$$

Tāpiri -4b ki te b.

$$-3b+58=17$$

Me tango 58 mai i ngā taha e rua o te whārite.

$$-3b=-41$$

Whakawehea ngā taha e rua ki te -3.

$$b=\frac{41}{3}$$

Whakaurua te \frac{41}{3} mō b ki a=-2b+29. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-2\times \frac{41}{3}+29$$

Whakareatia -2 ki te \frac{41}{3}.

$$a=-\frac{82}{3}+29$$

Tāpiri 29 ki te -\frac{82}{3}.

$$a=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{5}{3},b=\frac{41}{3}$$

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.

$$a+2b=29,2a+b=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\2&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}29\\17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}1&2\\2&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}29\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}29\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}29\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\times 2}&-\frac{2}{1-2\times 2}\\-\frac{2}{1-2\times 2}&\frac{1}{1-2\times 2}\end{matrix}\right)\left(\begin{matrix}29\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}29\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 29+\frac{2}{3}\times 17\\\frac{2}{3}\times 29-\frac{1}{3}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}\\\frac{41}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{5}{3},b=\frac{41}{3}$$

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.

$$a+2b=29,2a+b=17$$

Kia ōrite ai a a me 2a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2a+2\times 2b=2\times 29,2a+b=17$$

Whakarūnātia.

$$2a+4b=58,2a+b=17$$

Me tango 2a+b=17 mai i 2a+4b=58 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2a-2a+4b-b=58-17$$

Tāpiri 2a ki te -2a. Ka whakakore atu ngā kupu 2a me -2a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4b-b=58-17$$

Tāpiri 4b ki te -b.

$$3b=58-17$$

Tāpiri 58 ki te -17.

$$3b=41$$

Whakawehea ngā taha e rua ki te 3.

$$b=\frac{41}{3}$$

Whakaurua te \frac{41}{3} mō b ki 2a+b=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$2a+\frac{41}{3}=17$$

Me tango \frac{41}{3} mai i ngā taha e rua o te whārite.

$$2a=\frac{10}{3}$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{5}{3},b=\frac{41}{3}$$