Whakaoti i te {l}{4b+4c=-5}{4b+5c=-6}

$$\left. \begin{array} { l } { 4 b + 4 c = - 5 } \\ { 4 b + 5 c = - 6 } \end{array} \right.$$

Whakaoti mō b, c

$$b=-\frac{1}{4}=-0.25$$

$$c=-1$$

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.

$$4b+4c=-5,4b+5c=-6$$

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

$$4b+4c=-5$$

Me tango 4c mai i ngā taha e rua o te whārite.

$$4b=-4c-5$$

Whakawehea ngā taha e rua ki te 4.

$$b=\frac{1}{4}\left(-4c-5\right)$$

Whakareatia \frac{1}{4} ki te -4c-5.

$$b=-c-\frac{5}{4}$$

Whakakapia te -c-\frac{5}{4} mō te b ki tērā atu whārite, 4b+5c=-6.

$$4\left(-c-\frac{5}{4}\right)+5c=-6$$

Whakareatia 4 ki te -c-\frac{5}{4}.

$$-4c-5+5c=-6$$

Tāpiri -4c ki te 5c.

$$c-5=-6$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$c=-1$$

Whakaurua te -1 mō c ki b=-c-\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$b=-\left(-1\right)-\frac{5}{4}$$

Whakareatia -1 ki te -1.

$$b=1-\frac{5}{4}$$

Tāpiri -\frac{5}{4} ki te 1.

$$b=-\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$b=-\frac{1}{4},c=-1$$

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.

$$4b+4c=-5,4b+5c=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&4\\4&5\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&4\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&4\\4&5\end{matrix}\right))\left(\begin{matrix}4&4\\4&5\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}4&4\\4&5\end{matrix}\right))\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}4&4\\4&5\end{matrix}\right))\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}4&4\\4&5\end{matrix}\right))\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-4\times 4}&-\frac{4}{4\times 5-4\times 4}\\-\frac{4}{4\times 5-4\times 4}&\frac{4}{4\times 5-4\times 4}\end{matrix}\right)\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4}&-1\\-1&1\end{matrix}\right)\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4}\left(-5\right)-\left(-6\right)\\-\left(-5\right)-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa b me c.

$$b=-\frac{1}{4},c=-1$$

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.

$$4b+4c=-5,4b+5c=-6$$

Me tango 4b+5c=-6 mai i 4b+4c=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4b-4b+4c-5c=-5+6$$

Tāpiri 4b ki te -4b. Ka whakakore atu ngā kupu 4b me -4b, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4c-5c=-5+6$$

Tāpiri 4c ki te -5c.

$$-c=-5+6$$

Tāpiri -5 ki te 6.

$$-c=1$$

Whakawehea ngā taha e rua ki te -1.

$$c=-1$$

Whakaurua te -1 mō c ki 4b+5c=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$4b+5\left(-1\right)=-6$$

Whakareatia 5 ki te -1.

$$4b-5=-6$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$4b=-1$$

Whakawehea ngā taha e rua ki te 4.

$$b=-\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$b=-\frac{1}{4},c=-1$$