Whakaoti i te {l}{0=1-b+c}{0=-9+3b+c}

$$\left\{ \begin{array} { l } { 0 = 1 - b + c } \\ { 0 = - 9 + 3 b + c } \end{array} \right.$$

Whakaoti mō b, c

$$b = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$c = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

$$1-b+c=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-b+c=-1$$

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

$$-9+3b+c=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3b+c=9$$

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.

$$-b+c=-1,3b+c=9$$

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

$$-b+c=-1$$

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

$$-b=-c-1$$

Whakawehea ngā taha e rua ki te -1.

$$b=-\left(-c-1\right)$$

Whakareatia -1 ki te -c-1.

$$b=c+1$$

Whakakapia te c+1 mō te b ki tērā atu whārite, 3b+c=9.

$$3\left(c+1\right)+c=9$$

Whakareatia 3 ki te c+1.

$$3c+3+c=9$$

Tāpiri 3c ki te c.

$$4c+3=9$$

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

$$4c=6$$

Whakawehea ngā taha e rua ki te 4.

$$c=\frac{3}{2}$$

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

$$b=\frac{3}{2}+1$$

Tāpiri 1 ki te \frac{3}{2}.

$$b=\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$b=\frac{5}{2},c=\frac{3}{2}$$

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.

$$1-b+c=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-b+c=-1$$

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

$$-9+3b+c=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3b+c=9$$

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

$$-b+c=-1,3b+c=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{3}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-1\right)+\frac{1}{4}\times 9\\\frac{3}{4}\left(-1\right)+\frac{1}{4}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}b\\c\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2}\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa b me c.

$$b=\frac{5}{2},c=\frac{3}{2}$$

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.

$$1-b+c=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-b+c=-1$$

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

$$-9+3b+c=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3b+c=9$$

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.

$$-b+c=-1,3b+c=9$$

Me tango 3b+c=9 mai i -b+c=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-b-3b+c-c=-1-9$$

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.

$$-b-3b=-1-9$$

Tāpiri -b ki te -3b.

$$-4b=-1-9$$

Tāpiri -1 ki te -9.

$$-4b=-10$$

Whakawehea ngā taha e rua ki te -4.

$$b=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō b ki 3b+c=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$3\times \frac{5}{2}+c=9$$

Whakareatia 3 ki te \frac{5}{2}.

$$\frac{15}{2}+c=9$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$c=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$b=\frac{5}{2},c=\frac{3}{2}$$