Whakaoti i te {l}{0=1/3-b+c}{0=3+3b+c}

$$\left. \begin{array} { l } { 0 = \frac { 1 } { 3 } - b + c } \\ { 0 = 3 + 3 b + c } \end{array} \right.$$

Whakaoti mō b, c

$$b=-\frac{2}{3}\approx -0.666666667$$

$$c=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$\frac{1}{3}-b+c=0$$

Tangohia te \frac{1}{3} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3+3b+c=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3b+c=-3$$

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.

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

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

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

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

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

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te -c-\frac{1}{3}.

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

Whakakapia te c+\frac{1}{3} mō te b ki tērā atu whārite, 3b+c=-3.

$$3\left(c+\frac{1}{3}\right)+c=-3$$

Whakareatia 3 ki te c+\frac{1}{3}.

$$3c+1+c=-3$$

Tāpiri 3c ki te c.

$$4c+1=-3$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$4c=-4$$

Whakawehea ngā taha e rua ki te 4.

$$c=-1$$

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

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

Tāpiri \frac{1}{3} ki te -1.

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

Kua oti te pūnaha te whakatau.

$$b=-\frac{2}{3},c=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$\frac{1}{3}-b+c=0$$

Tangohia te \frac{1}{3} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3+3b+c=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3b+c=-3$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}b\\c\end{matrix}\right)=inverse(\left(\begin{matrix}-1&1\\3&1\end{matrix}\right))\left(\begin{matrix}-\frac{1}{3}\\-3\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{3}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-\frac{1}{3}\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-\frac{1}{3}\right)+\frac{1}{4}\left(-3\right)\\\frac{3}{4}\left(-\frac{1}{3}\right)+\frac{1}{4}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa b me c.

$$b=-\frac{2}{3},c=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$\frac{1}{3}-b+c=0$$

Tangohia te \frac{1}{3} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3+3b+c=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3b+c=-3$$

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.

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

Me tango 3b+c=-3 mai i -b+c=-\frac{1}{3} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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

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

Tāpiri -b ki te -3b.

$$-4b=-\frac{1}{3}+3$$

Tāpiri -\frac{1}{3} ki te 3.

$$-4b=\frac{8}{3}$$

Whakawehea ngā taha e rua ki te -4.

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

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

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

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

$$-2+c=-3$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$c=-1$$

Kua oti te pūnaha te whakatau.

$$b=-\frac{2}{3},c=-1$$