Whakaoti i te {l}{3a+b=9}{a+b=3}

$$\left. \begin{array} { l } { 3 a + b = 9 } \\ { a + b = 3 } \end{array} \right.$$

Whakaoti mō a, b

$$a=3$$

$$b=0$$

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.

$$3a+b=9,a+b=3$$

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

$$3a+b=9$$

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

$$3a=-b+9$$

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{1}{3}\left(-b+9\right)$$

Whakareatia \frac{1}{3} ki te -b+9.

$$a=-\frac{1}{3}b+3$$

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

$$-\frac{1}{3}b+3+b=3$$

Tāpiri -\frac{b}{3} ki te b.

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

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

$$\frac{2}{3}b=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=0$$

Whakaurua te 0 mō b ki a=-\frac{1}{3}b+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,b=0$$

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.

$$3a+b=9,a+b=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}9\\3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}3&1\\1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}9\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}9\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&1\end{matrix}\right))\left(\begin{matrix}9\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-1}&-\frac{1}{3-1}\\-\frac{1}{3-1}&\frac{3}{3-1}\end{matrix}\right)\left(\begin{matrix}9\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\-\frac{1}{2}&\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}9\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 9-\frac{1}{2}\times 3\\-\frac{1}{2}\times 9+\frac{3}{2}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=3,b=0$$

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.

$$3a+b=9,a+b=3$$

Me tango a+b=3 mai i 3a+b=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3a-a+b-b=9-3$$

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

$$3a-a=9-3$$

Tāpiri 3a ki te -a.

$$2a=9-3$$

Tāpiri 9 ki te -3.

$$2a=6$$

Whakawehea ngā taha e rua ki te 2.

$$a=3$$

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

$$3+b=3$$

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

$$b=0$$

Kua oti te pūnaha te whakatau.

$$a=3,b=0$$