Whakaoti i te {l}{s-t=3}{s/3+t/2=6}

$$\left. \begin{array} { l } { s - t = 3 } \\ { \frac { s } { 3 } + \frac { t } { 2 } = 6 } \end{array} \right.$$

Whakaoti mō s, t

$$t=6$$

$$s=9$$

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.

$$s-t=3,\frac{1}{3}s+\frac{1}{2}t=6$$

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

$$s-t=3$$

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

$$s=t+3$$

Whakakapia te t+3 mō te s ki tērā atu whārite, \frac{1}{3}s+\frac{1}{2}t=6.

$$\frac{1}{3}\left(t+3\right)+\frac{1}{2}t=6$$

Whakareatia \frac{1}{3} ki te t+3.

$$\frac{1}{3}t+1+\frac{1}{2}t=6$$

Tāpiri \frac{t}{3} ki te \frac{t}{2}.

$$\frac{5}{6}t+1=6$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$\frac{5}{6}t=5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$t=6$$

Whakaurua te 6 mō t ki s=t+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō s hāngai tonu.

$$s=6+3$$

Tāpiri 3 ki te 6.

$$s=9$$

Kua oti te pūnaha te whakatau.

$$s=9,t=6$$

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.

$$s-t=3,\frac{1}{3}s+\frac{1}{2}t=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}3\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1&-1\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}s\\t\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}3\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}s\\t\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}3\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}s\\t\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}3\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{2}}{\frac{1}{2}-\left(-\frac{1}{3}\right)}&-\frac{-1}{\frac{1}{2}-\left(-\frac{1}{3}\right)}\\-\frac{\frac{1}{3}}{\frac{1}{2}-\left(-\frac{1}{3}\right)}&\frac{1}{\frac{1}{2}-\left(-\frac{1}{3}\right)}\end{matrix}\right)\left(\begin{matrix}3\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&\frac{6}{5}\\-\frac{2}{5}&\frac{6}{5}\end{matrix}\right)\left(\begin{matrix}3\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 3+\frac{6}{5}\times 6\\-\frac{2}{5}\times 3+\frac{6}{5}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}9\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa s me t.

$$s=9,t=6$$

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.

$$s-t=3,\frac{1}{3}s+\frac{1}{2}t=6$$

Kia ōrite ai a s me \frac{s}{3}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{3} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$\frac{1}{3}s+\frac{1}{3}\left(-1\right)t=\frac{1}{3}\times 3,\frac{1}{3}s+\frac{1}{2}t=6$$

Whakarūnātia.

$$\frac{1}{3}s-\frac{1}{3}t=1,\frac{1}{3}s+\frac{1}{2}t=6$$

Me tango \frac{1}{3}s+\frac{1}{2}t=6 mai i \frac{1}{3}s-\frac{1}{3}t=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{3}s-\frac{1}{3}s-\frac{1}{3}t-\frac{1}{2}t=1-6$$

Tāpiri \frac{s}{3} ki te -\frac{s}{3}. Ka whakakore atu ngā kupu \frac{s}{3} me -\frac{s}{3}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-\frac{1}{3}t-\frac{1}{2}t=1-6$$

Tāpiri -\frac{t}{3} ki te -\frac{t}{2}.

$$-\frac{5}{6}t=1-6$$

Tāpiri 1 ki te -6.

$$-\frac{5}{6}t=-5$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$t=6$$

Whakaurua te 6 mō t ki \frac{1}{3}s+\frac{1}{2}t=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō s hāngai tonu.

$$\frac{1}{3}s+\frac{1}{2}\times 6=6$$

Whakareatia \frac{1}{2} ki te 6.

$$\frac{1}{3}s+3=6$$

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

$$\frac{1}{3}s=3$$

Me whakarea ngā taha e rua ki te 3.

$$s=9$$

Kua oti te pūnaha te whakatau.

$$s=9,t=6$$