Whakaoti i te {l}{6+2a+b=0}{24-4a+b=0}

$$\left\{ \begin{array} { l } { 6 + 2 a + b = 0 } \\ { 24 - 4 a + b = 0 } \end{array} \right.$$

Whakaoti mō a, b

$$a=3$$

$$b=-12$$

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.

$$2a+b+6=0,-4a+b+24=0$$

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

$$2a+b+6=0$$

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

$$2a+b=-6$$

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

$$2a=-b-6$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{1}{2}\left(-b-6\right)$$

Whakareatia \frac{1}{2} ki te -b-6.

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

Whakakapia te -\frac{b}{2}-3 mō te a ki tērā atu whārite, -4a+b+24=0.

$$-4\left(-\frac{1}{2}b-3\right)+b+24=0$$

Whakareatia -4 ki te -\frac{b}{2}-3.

$$2b+12+b+24=0$$

Tāpiri 2b ki te b.

$$3b+12+24=0$$

Tāpiri 12 ki te 24.

$$3b+36=0$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$3b=-36$$

Whakawehea ngā taha e rua ki te 3.

$$b=-12$$

Whakaurua te -12 mō b ki a=-\frac{1}{2}b-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{1}{2}\left(-12\right)-3$$

Whakareatia -\frac{1}{2} ki te -12.

$$a=6-3$$

Tāpiri -3 ki te 6.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,b=-12$$

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.

$$2a+b+6=0,-4a+b+24=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\-4&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-6\\-24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\-4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\-4&1\end{matrix}\right))\left(\begin{matrix}2&1\\-4&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&1\end{matrix}\right))\left(\begin{matrix}-6\\-24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&1\end{matrix}\right))\left(\begin{matrix}-6\\-24\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&1\end{matrix}\right))\left(\begin{matrix}-6\\-24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-4\right)}&-\frac{1}{2-\left(-4\right)}\\-\frac{-4}{2-\left(-4\right)}&\frac{2}{2-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{1}{6}\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-6\\-24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\left(-6\right)-\frac{1}{6}\left(-24\right)\\\frac{2}{3}\left(-6\right)+\frac{1}{3}\left(-24\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}3\\-12\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=3,b=-12$$

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.

$$2a+b+6=0,-4a+b+24=0$$

Me tango -4a+b+24=0 mai i 2a+b+6=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2a+4a+b-b+6-24=0$$

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.

$$2a+4a+6-24=0$$

Tāpiri 2a ki te 4a.

$$6a+6-24=0$$

Tāpiri 6 ki te -24.

$$6a-18=0$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$6a=18$$

Whakawehea ngā taha e rua ki te 6.

$$a=3$$

Whakaurua te 3 mō a ki -4a+b+24=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$-4\times 3+b+24=0$$

Whakareatia -4 ki te 3.

$$-12+b+24=0$$

Tāpiri -12 ki te 24.

$$b+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$b=-12$$

Kua oti te pūnaha te whakatau.

$$a=3,b=-12$$