Whakaoti i te {l}{3u+z=15}{u+2z=10}

$$\left. \begin{array} { l } { 3 u + z = 15 } \\ { u + 2 z = 10 } \end{array} \right.$$

Whakaoti mō u, z

$$z=3$$

$$u=4$$

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.

$$3u+z=15,u+2z=10$$

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

$$3u+z=15$$

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

$$3u=-z+15$$

Whakawehea ngā taha e rua ki te 3.

$$u=\frac{1}{3}\left(-z+15\right)$$

Whakareatia \frac{1}{3} ki te -z+15.

$$u=-\frac{1}{3}z+5$$

Whakakapia te -\frac{z}{3}+5 mō te u ki tērā atu whārite, u+2z=10.

$$-\frac{1}{3}z+5+2z=10$$

Tāpiri -\frac{z}{3} ki te 2z.

$$\frac{5}{3}z+5=10$$

Me tango 5 mai i ngā taha e rua o te whārite.

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

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.

$$z=3$$

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

$$u=-\frac{1}{3}\times 3+5$$

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

$$u=-1+5$$

Tāpiri 5 ki te -1.

$$u=4$$

Kua oti te pūnaha te whakatau.

$$u=4,z=3$$

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.

$$3u+z=15,u+2z=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\1&2\end{matrix}\right)\left(\begin{matrix}u\\z\end{matrix}\right)=\left(\begin{matrix}15\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\1&2\end{matrix}\right))\left(\begin{matrix}3&1\\1&2\end{matrix}\right)\left(\begin{matrix}u\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&2\end{matrix}\right))\left(\begin{matrix}15\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}u\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&2\end{matrix}\right))\left(\begin{matrix}15\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}u\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&2\end{matrix}\right))\left(\begin{matrix}15\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}u\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-1}&-\frac{1}{3\times 2-1}\\-\frac{1}{3\times 2-1}&\frac{3}{3\times 2-1}\end{matrix}\right)\left(\begin{matrix}15\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\-\frac{1}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}15\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}u\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 15-\frac{1}{5}\times 10\\-\frac{1}{5}\times 15+\frac{3}{5}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\z\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa u me z.

$$u=4,z=3$$

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.

$$3u+z=15,u+2z=10$$

Kia ōrite ai a 3u me u, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3u+z=15,3u+3\times 2z=3\times 10$$

Whakarūnātia.

$$3u+z=15,3u+6z=30$$

Me tango 3u+6z=30 mai i 3u+z=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3u-3u+z-6z=15-30$$

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.

$$z-6z=15-30$$

Tāpiri z ki te -6z.

$$-5z=15-30$$

Tāpiri 15 ki te -30.

$$-5z=-15$$

Whakawehea ngā taha e rua ki te -5.

$$z=3$$

Whakaurua te 3 mō z ki u+2z=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$u+2\times 3=10$$

Whakareatia 2 ki te 3.

$$u+6=10$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$u=4$$

Kua oti te pūnaha te whakatau.

$$u=4,z=3$$