Whakaoti i te {l}{2a+3b=4}{-2a+3b=-16}

$$\left\{ \begin{array} { l } { 2 a + 3 b = 4 } \\ { - 2 a + 3 b = - 16 } \end{array} \right.$$

Whakaoti mō a, b

$$a=5$$

$$b=-2$$

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.

$$2a+3b=4,-2a+3b=-16$$

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

$$2a+3b=4$$

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

$$2a=-3b+4$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -3b+4.

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

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

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

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

$$3b-4+3b=-16$$

Tāpiri 3b ki te 3b.

$$6b-4=-16$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$6b=-12$$

Whakawehea ngā taha e rua ki te 6.

$$b=-2$$

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

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

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

$$a=3+2$$

Tāpiri 2 ki te 3.

$$a=5$$

Kua oti te pūnaha te whakatau.

$$a=5,b=-2$$

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.

$$2a+3b=4,-2a+3b=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{4}\\\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}4\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 4-\frac{1}{4}\left(-16\right)\\\frac{1}{6}\times 4+\frac{1}{6}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=5,b=-2$$

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.

$$2a+3b=4,-2a+3b=-16$$

Me tango -2a+3b=-16 mai i 2a+3b=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2a+2a+3b-3b=4+16$$

Tāpiri 3b ki te -3b. Ka whakakore atu ngā kupu 3b me -3b, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2a+2a=4+16$$

Tāpiri 2a ki te 2a.

$$4a=4+16$$

Tāpiri 4 ki te 16.

$$4a=20$$

Whakawehea ngā taha e rua ki te 4.

$$a=5$$

Whakaurua te 5 mō a ki -2a+3b=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$-2\times 5+3b=-16$$

Whakareatia -2 ki te 5.

$$-10+3b=-16$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$3b=-6$$

Whakawehea ngā taha e rua ki te 3.

$$b=-2$$

Kua oti te pūnaha te whakatau.

$$a=5,b=-2$$