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

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

Whakaoti mō a, b

$$a=4$$

$$b=-3$$

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.

$$-1+a+b=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$a+b=1$$

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

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

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

$$3a+b=9$$

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.

$$a+b=1,3a+b=9$$

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

$$a+b=1$$

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

$$a=-b+1$$

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

$$3\left(-b+1\right)+b=9$$

Whakareatia 3 ki te -b+1.

$$-3b+3+b=9$$

Tāpiri -3b ki te b.

$$-2b+3=9$$

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

$$-2b=6$$

Whakawehea ngā taha e rua ki te -2.

$$b=-3$$

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

$$a=-\left(-3\right)+1$$

Whakareatia -1 ki te -3.

$$a=3+1$$

Tāpiri 1 ki te 3.

$$a=4$$

Kua oti te pūnaha te whakatau.

$$a=4,b=-3$$

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.

$$-1+a+b=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$a+b=1$$

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

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

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

$$3a+b=9$$

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

$$a+b=1,3a+b=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\\frac{3}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}1\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}+\frac{1}{2}\times 9\\\frac{3}{2}-\frac{1}{2}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=4,b=-3$$

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.

$$-1+a+b=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$a+b=1$$

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

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

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

$$3a+b=9$$

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.

$$a+b=1,3a+b=9$$

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

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

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.

$$a-3a=1-9$$

Tāpiri a ki te -3a.

$$-2a=1-9$$

Tāpiri 1 ki te -9.

$$-2a=-8$$

Whakawehea ngā taha e rua ki te -2.

$$a=4$$

Whakaurua te 4 mō a ki 3a+b=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$3\times 4+b=9$$

Whakareatia 3 ki te 4.

$$12+b=9$$

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

$$b=-3$$

Kua oti te pūnaha te whakatau.

$$a=4,b=-3$$