Whakaoti i te {l}{2a=3b}{7a+2b=200}

$$\left\{ \begin{array} { l } { 2 a = 3 b } \\ { 7 a + 2 b = 200 } \end{array} \right.$$

Whakaoti mō a, b

$$a=24$$

$$b=16$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3b mai i ngā taha e rua.

$$2a-3b=0$$

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.

$$2a-3b=0,7a+2b=200$$

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

$$2a-3b=0$$

Me tāpiri 3b ki ngā taha e rua o te whārite.

$$2a=3b$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{1}{2}\times 3b$$

Whakareatia \frac{1}{2} ki te 3b.

$$a=\frac{3}{2}b$$

Whakakapia te \frac{3b}{2} mō te a ki tērā atu whārite, 7a+2b=200.

$$7\times \frac{3}{2}b+2b=200$$

Whakareatia 7 ki te \frac{3b}{2}.

$$\frac{21}{2}b+2b=200$$

Tāpiri \frac{21b}{2} ki te 2b.

$$\frac{25}{2}b=200$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=16$$

Whakaurua te 16 mō b ki a=\frac{3}{2}b. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=\frac{3}{2}\times 16$$

Whakareatia \frac{3}{2} ki te 16.

$$a=24$$

Kua oti te pūnaha te whakatau.

$$a=24,b=16$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3b mai i ngā taha e rua.

$$2a-3b=0$$

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

$$2a-3b=0,7a+2b=200$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\7&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}0\\200\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\7&2\end{matrix}\right))\left(\begin{matrix}2&-3\\7&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\7&2\end{matrix}\right))\left(\begin{matrix}0\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\7&2\end{matrix}\right))\left(\begin{matrix}0\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\7&2\end{matrix}\right))\left(\begin{matrix}0\\200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-\left(-3\times 7\right)}&-\frac{-3}{2\times 2-\left(-3\times 7\right)}\\-\frac{7}{2\times 2-\left(-3\times 7\right)}&\frac{2}{2\times 2-\left(-3\times 7\right)}\end{matrix}\right)\left(\begin{matrix}0\\200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{25}&\frac{3}{25}\\-\frac{7}{25}&\frac{2}{25}\end{matrix}\right)\left(\begin{matrix}0\\200\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{25}\times 200\\\frac{2}{25}\times 200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}24\\16\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=24,b=16$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3b mai i ngā taha e rua.

$$2a-3b=0$$

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.

$$2a-3b=0,7a+2b=200$$

Kia ōrite ai a 2a me 7a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$7\times 2a+7\left(-3\right)b=0,2\times 7a+2\times 2b=2\times 200$$

Whakarūnātia.

$$14a-21b=0,14a+4b=400$$

Me tango 14a+4b=400 mai i 14a-21b=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14a-14a-21b-4b=-400$$

Tāpiri 14a ki te -14a. Ka whakakore atu ngā kupu 14a me -14a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-21b-4b=-400$$

Tāpiri -21b ki te -4b.

$$-25b=-400$$

Whakawehea ngā taha e rua ki te -25.

$$b=16$$

Whakaurua te 16 mō b ki 7a+2b=200. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$7a+2\times 16=200$$

Whakareatia 2 ki te 16.

$$7a+32=200$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$7a=168$$

Whakawehea ngā taha e rua ki te 7.

$$a=24$$

Kua oti te pūnaha te whakatau.

$$a=24,b=16$$