Whakaoti i te {l}{k/L=100}{5k+50L=110}

$$\left. \begin{array} { l } { \frac { k } { L } = 100 } \\ { 5 k + 50 L = 110 } \end{array} \right.$$

Whakaoti mō k, L

$$k=20$$

$$L=\frac{1}{5}=0.2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe L ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te L.

$$k=100L$$

Whakakapia te 100L mō te k ki tērā atu whārite, 5k+50L=110.

$$5\times 100L+50L=110$$

Whakareatia 5 ki te 100L.

$$500L+50L=110$$

Tāpiri 500L ki te 50L.

$$550L=110$$

Whakawehea ngā taha e rua ki te 550.

$$L=\frac{1}{5}$$

Whakaurua te \frac{1}{5} mō L ki k=100L. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$k=100\times \frac{1}{5}$$

Whakareatia 100 ki te \frac{1}{5}.

$$k=20$$

Kua oti te pūnaha te whakatau.

$$k=20,L=\frac{1}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe L ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te L.

$$k=100L$$

Tangohia te 100L mai i ngā taha e rua.

$$k-100L=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$k-100L=0,5k+50L=110$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-100\\5&50\end{matrix}\right)\left(\begin{matrix}k\\L\end{matrix}\right)=\left(\begin{matrix}0\\110\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-100\\5&50\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-100\\5&50\end{matrix}\right))\left(\begin{matrix}1&-100\\5&50\end{matrix}\right)\left(\begin{matrix}k\\L\end{matrix}\right)=inverse(\left(\begin{matrix}1&-100\\5&50\end{matrix}\right))\left(\begin{matrix}0\\110\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}k\\L\end{matrix}\right)=inverse(\left(\begin{matrix}1&-100\\5&50\end{matrix}\right))\left(\begin{matrix}0\\110\end{matrix}\right)$$

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

$$\left(\begin{matrix}k\\L\end{matrix}\right)=inverse(\left(\begin{matrix}1&-100\\5&50\end{matrix}\right))\left(\begin{matrix}0\\110\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}k\\L\end{matrix}\right)=\left(\begin{matrix}\frac{50}{50-\left(-100\times 5\right)}&-\frac{-100}{50-\left(-100\times 5\right)}\\-\frac{5}{50-\left(-100\times 5\right)}&\frac{1}{50-\left(-100\times 5\right)}\end{matrix}\right)\left(\begin{matrix}0\\110\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\L\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{2}{11}\\-\frac{1}{110}&\frac{1}{550}\end{matrix}\right)\left(\begin{matrix}0\\110\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}k\\L\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 110\\\frac{1}{550}\times 110\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\L\end{matrix}\right)=\left(\begin{matrix}20\\\frac{1}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa k me L.

$$k=20,L=\frac{1}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe L ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te L.

$$k=100L$$

Tangohia te 100L mai i ngā taha e rua.

$$k-100L=0$$

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.

$$k-100L=0,5k+50L=110$$

Kia ōrite ai a k me 5k, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5k+5\left(-100\right)L=0,5k+50L=110$$

Whakarūnātia.

$$5k-500L=0,5k+50L=110$$

Me tango 5k+50L=110 mai i 5k-500L=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5k-5k-500L-50L=-110$$

Tāpiri 5k ki te -5k. Ka whakakore atu ngā kupu 5k me -5k, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-500L-50L=-110$$

Tāpiri -500L ki te -50L.

$$-550L=-110$$

Whakawehea ngā taha e rua ki te -550.

$$L=\frac{1}{5}$$

Whakaurua te \frac{1}{5} mō L ki 5k+50L=110. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$5k+50\times \frac{1}{5}=110$$

Whakareatia 50 ki te \frac{1}{5}.

$$5k+10=110$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$5k=100$$

Whakawehea ngā taha e rua ki te 5.

$$k=20$$

Kua oti te pūnaha te whakatau.

$$k=20,L=\frac{1}{5}$$