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

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

Whakaoti mō a, b

$$a=\frac{1}{2}=0.5$$

$$b=\frac{1}{2}=0.5$$

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.

$$a+b=1$$

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

$$3a-b=1$$

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.

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

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=1.

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

Whakareatia 3 ki te -b+1.

$$-3b+3-b=1$$

Tāpiri -3b ki te -b.

$$-4b+3=1$$

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

$$-4b=-2$$

Whakawehea ngā taha e rua ki te -4.

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

Whakaurua te \frac{1}{2} 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.

$$a=-\frac{1}{2}+1$$

Tāpiri 1 ki te -\frac{1}{2}.

$$a=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{1}{2},b=\frac{1}{2}$$

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.

$$a+b=1$$

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

$$3a-b=1$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=\frac{1}{2},b=\frac{1}{2}$$

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.

$$a+b=1$$

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

$$3a-b=1$$

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.

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

Kia ōrite ai a a me 3a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

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

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

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

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

$$3b+b=3-1$$

Tāpiri 3b ki te b.

$$4b=3-1$$

Tāpiri 3 ki te -1.

$$4b=2$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$3a-\frac{1}{2}=1$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{1}{2},b=\frac{1}{2}$$