Whakaoti i te {l}{3a+b=-3}{2a-b=-1}

$$\left. \begin{array} { l } { 3 a + b = - 3 } \\ { 2 a - b = - 1 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-\frac{4}{5}=-0.8$$

$$b=-\frac{3}{5}=-0.6$$

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.

$$3a+b=-3,2a-b=-1$$

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

$$3a+b=-3$$

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

$$3a=-b-3$$

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{1}{3}\left(-b-3\right)$$

Whakareatia \frac{1}{3} ki te -b-3.

$$a=-\frac{1}{3}b-1$$

Whakakapia te -\frac{b}{3}-1 mō te a ki tērā atu whārite, 2a-b=-1.

$$2\left(-\frac{1}{3}b-1\right)-b=-1$$

Whakareatia 2 ki te -\frac{b}{3}-1.

$$-\frac{2}{3}b-2-b=-1$$

Tāpiri -\frac{2b}{3} ki te -b.

$$-\frac{5}{3}b-2=-1$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-\frac{5}{3}b=1$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-\frac{3}{5}$$

Whakaurua te -\frac{3}{5} mō b ki a=-\frac{1}{3}b-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{1}{3}\left(-\frac{3}{5}\right)-1$$

Whakareatia -\frac{1}{3} ki te -\frac{3}{5} 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.

$$a=\frac{1}{5}-1$$

Tāpiri -1 ki te \frac{1}{5}.

$$a=-\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{4}{5},b=-\frac{3}{5}$$

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.

$$3a+b=-3,2a-b=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\2&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-3\right)+\frac{1}{5}\left(-1\right)\\\frac{2}{5}\left(-3\right)-\frac{3}{5}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=-\frac{4}{5},b=-\frac{3}{5}$$

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.

$$3a+b=-3,2a-b=-1$$

Kia ōrite ai a 3a 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 3.

$$2\times 3a+2b=2\left(-3\right),3\times 2a+3\left(-1\right)b=3\left(-1\right)$$

Whakarūnātia.

$$6a+2b=-6,6a-3b=-3$$

Me tango 6a-3b=-3 mai i 6a+2b=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6a-6a+2b+3b=-6+3$$

Tāpiri 6a ki te -6a. Ka whakakore atu ngā kupu 6a me -6a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2b+3b=-6+3$$

Tāpiri 2b ki te 3b.

$$5b=-6+3$$

Tāpiri -6 ki te 3.

$$5b=-3$$

Whakawehea ngā taha e rua ki te 5.

$$b=-\frac{3}{5}$$

Whakaurua te -\frac{3}{5} mō b ki 2a-b=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$2a-\left(-\frac{3}{5}\right)=-1$$

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$2a=-\frac{8}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$a=-\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{4}{5},b=-\frac{3}{5}$$