Whakaoti i te {l}{3t-2u=7}{9t-5u=18}

$$\left. \begin{array} { l } { 3 t - 2 u = 7 } \\ { 9 t - 5 u = 18 } \end{array} \right.$$

Whakaoti mō t, u

$$t=\frac{1}{3}\approx 0.333333333$$

$$u=-3$$

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.

$$3t-2u=7,9t-5u=18$$

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

$$3t-2u=7$$

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

$$3t=2u+7$$

Whakawehea ngā taha e rua ki te 3.

$$t=\frac{1}{3}\left(2u+7\right)$$

Whakareatia \frac{1}{3} ki te 2u+7.

$$t=\frac{2}{3}u+\frac{7}{3}$$

Whakakapia te \frac{2u+7}{3} mō te t ki tērā atu whārite, 9t-5u=18.

$$9\left(\frac{2}{3}u+\frac{7}{3}\right)-5u=18$$

Whakareatia 9 ki te \frac{2u+7}{3}.

$$6u+21-5u=18$$

Tāpiri 6u ki te -5u.

$$u+21=18$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$u=-3$$

Whakaurua te -3 mō u ki t=\frac{2}{3}u+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō t hāngai tonu.

$$t=\frac{2}{3}\left(-3\right)+\frac{7}{3}$$

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

$$t=-2+\frac{7}{3}$$

Tāpiri \frac{7}{3} ki te -2.

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

Kua oti te pūnaha te whakatau.

$$t=\frac{1}{3},u=-3$$

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.

$$3t-2u=7,9t-5u=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\9&-5\end{matrix}\right)\left(\begin{matrix}t\\u\end{matrix}\right)=\left(\begin{matrix}7\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\9&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\9&-5\end{matrix}\right))\left(\begin{matrix}3&-2\\9&-5\end{matrix}\right)\left(\begin{matrix}t\\u\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\9&-5\end{matrix}\right))\left(\begin{matrix}7\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}t\\u\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\9&-5\end{matrix}\right))\left(\begin{matrix}7\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}t\\u\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\9&-5\end{matrix}\right))\left(\begin{matrix}7\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}t\\u\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3\left(-5\right)-\left(-2\times 9\right)}&-\frac{-2}{3\left(-5\right)-\left(-2\times 9\right)}\\-\frac{9}{3\left(-5\right)-\left(-2\times 9\right)}&\frac{3}{3\left(-5\right)-\left(-2\times 9\right)}\end{matrix}\right)\left(\begin{matrix}7\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}t\\u\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3}&\frac{2}{3}\\-3&1\end{matrix}\right)\left(\begin{matrix}7\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}t\\u\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3}\times 7+\frac{2}{3}\times 18\\-3\times 7+18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}t\\u\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa t me u.

$$t=\frac{1}{3},u=-3$$

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.

$$3t-2u=7,9t-5u=18$$

Kia ōrite ai a 3t me 9t, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$9\times 3t+9\left(-2\right)u=9\times 7,3\times 9t+3\left(-5\right)u=3\times 18$$

Whakarūnātia.

$$27t-18u=63,27t-15u=54$$

Me tango 27t-15u=54 mai i 27t-18u=63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27t-27t-18u+15u=63-54$$

Tāpiri 27t ki te -27t. Ka whakakore atu ngā kupu 27t me -27t, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-18u+15u=63-54$$

Tāpiri -18u ki te 15u.

$$-3u=63-54$$

Tāpiri 63 ki te -54.

$$-3u=9$$

Whakawehea ngā taha e rua ki te -3.

$$u=-3$$

Whakaurua te -3 mō u ki 9t-5u=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō t hāngai tonu.

$$9t-5\left(-3\right)=18$$

Whakareatia -5 ki te -3.

$$9t+15=18$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$9t=3$$

Whakawehea ngā taha e rua ki te 9.

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

Kua oti te pūnaha te whakatau.

$$t=\frac{1}{3},u=-3$$