Whakaoti i te {r}{u-30v=-65}{-3u+80v=165}

$$\left\{ \begin{array} { r } { u - 30 v = - 65 } \\ { - 3 u + 80 v = 165 } \end{array} \right.$$

Whakaoti mō u, v

$$u=25$$

$$v=3$$

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.

$$u-30v=-65,-3u+80v=165$$

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

$$u-30v=-65$$

Me tāpiri 30v ki ngā taha e rua o te whārite.

$$u=30v-65$$

Whakakapia te 30v-65 mō te u ki tērā atu whārite, -3u+80v=165.

$$-3\left(30v-65\right)+80v=165$$

Whakareatia -3 ki te 30v-65.

$$-90v+195+80v=165$$

Tāpiri -90v ki te 80v.

$$-10v+195=165$$

Me tango 195 mai i ngā taha e rua o te whārite.

$$-10v=-30$$

Whakawehea ngā taha e rua ki te -10.

$$v=3$$

Whakaurua te 3 mō v ki u=30v-65. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$u=30\times 3-65$$

Whakareatia 30 ki te 3.

$$u=90-65$$

Tāpiri -65 ki te 90.

$$u=25$$

Kua oti te pūnaha te whakatau.

$$u=25,v=3$$

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.

$$u-30v=-65,-3u+80v=165$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-30\\-3&80\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-65\\165\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-30\\-3&80\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-30\\-3&80\end{matrix}\right))\left(\begin{matrix}1&-30\\-3&80\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}1&-30\\-3&80\end{matrix}\right))\left(\begin{matrix}-65\\165\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}1&-30\\-3&80\end{matrix}\right))\left(\begin{matrix}-65\\165\end{matrix}\right)$$

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

$$\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}1&-30\\-3&80\end{matrix}\right))\left(\begin{matrix}-65\\165\end{matrix}\right)$$

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.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}\frac{80}{80-\left(-30\left(-3\right)\right)}&-\frac{-30}{80-\left(-30\left(-3\right)\right)}\\-\frac{-3}{80-\left(-30\left(-3\right)\right)}&\frac{1}{80-\left(-30\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}-65\\165\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-8&-3\\-\frac{3}{10}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}-65\\165\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-8\left(-65\right)-3\times 165\\-\frac{3}{10}\left(-65\right)-\frac{1}{10}\times 165\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}25\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa u me v.

$$u=25,v=3$$

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.

$$u-30v=-65,-3u+80v=165$$

Kia ōrite ai a u me -3u, 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.

$$-3u-3\left(-30\right)v=-3\left(-65\right),-3u+80v=165$$

Whakarūnātia.

$$-3u+90v=195,-3u+80v=165$$

Me tango -3u+80v=165 mai i -3u+90v=195 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3u+3u+90v-80v=195-165$$

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

$$90v-80v=195-165$$

Tāpiri 90v ki te -80v.

$$10v=195-165$$

Tāpiri 195 ki te -165.

$$10v=30$$

Whakawehea ngā taha e rua ki te 10.

$$v=3$$

Whakaurua te 3 mō v ki -3u+80v=165. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$-3u+80\times 3=165$$

Whakareatia 80 ki te 3.

$$-3u+240=165$$

Me tango 240 mai i ngā taha e rua o te whārite.

$$-3u=-75$$

Whakawehea ngā taha e rua ki te -3.

$$u=25$$

Kua oti te pūnaha te whakatau.

$$u=25,v=3$$