Whakaoti i te {l}{2a+3b=0}{2a+5b=16}

$$\left. \begin{array} { l } { 2 a + 3 b = 0 } \\ { 2 a + 5 b = 16 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-12$$

$$b=8$$

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.

$$2a+3b=0,2a+5b=16$$

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 tango 3b mai i ngā taha e rua o te whārite.

$$2a=-3b$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{1}{2}\left(-3\right)b$$

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

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

Whakakapia te -\frac{3b}{2} mō te a ki tērā atu whārite, 2a+5b=16.

$$2\left(-\frac{3}{2}\right)b+5b=16$$

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

$$-3b+5b=16$$

Tāpiri -3b ki te 5b.

$$2b=16$$

Whakawehea ngā taha e rua ki te 2.

$$b=8$$

Whakaurua te 8 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.

$$a=-\frac{3}{2}\times 8$$

Whakareatia -\frac{3}{2} ki te 8.

$$a=-12$$

Kua oti te pūnaha te whakatau.

$$a=-12,b=8$$

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.

$$2a+3b=0,2a+5b=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\2&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\2&5\end{matrix}\right))\left(\begin{matrix}2&3\\2&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\2&5\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\2&5\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\2&5\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-3\times 2}&-\frac{3}{2\times 5-3\times 2}\\-\frac{2}{2\times 5-3\times 2}&\frac{2}{2\times 5-3\times 2}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4}&-\frac{3}{4}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4}\times 16\\\frac{1}{2}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-12\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-12,b=8$$

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.

$$2a+3b=0,2a+5b=16$$

Me tango 2a+5b=16 mai i 2a+3b=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2a-2a+3b-5b=-16$$

Tāpiri 2a ki te -2a. Ka whakakore atu ngā kupu 2a me -2a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3b-5b=-16$$

Tāpiri 3b ki te -5b.

$$-2b=-16$$

Whakawehea ngā taha e rua ki te -2.

$$b=8$$

Whakaurua te 8 mō b ki 2a+5b=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$2a+5\times 8=16$$

Whakareatia 5 ki te 8.

$$2a+40=16$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$2a=-24$$

Whakawehea ngā taha e rua ki te 2.

$$a=-12$$

Kua oti te pūnaha te whakatau.

$$a=-12,b=8$$