Whakaoti i te {l}{a+5b=2}{a-2b=1}

$$\left\{ \begin{array} { l } { a + 5 b = 2 } \\ { a - 2 b = 1 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{9}{7} = 1\frac{2}{7} \approx 1.285714286$$

$$b=\frac{1}{7}\approx 0.142857143$$

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.

$$a+5b=2,a-2b=1$$

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

$$a+5b=2$$

Me tango 5b mai i ngā taha e rua o te whārite.

$$a=-5b+2$$

Whakakapia te -5b+2 mō te a ki tērā atu whārite, a-2b=1.

$$-5b+2-2b=1$$

Tāpiri -5b ki te -2b.

$$-7b+2=1$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-7b=-1$$

Whakawehea ngā taha e rua ki te -7.

$$b=\frac{1}{7}$$

Whakaurua te \frac{1}{7} mō b ki a=-5b+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-5\times \frac{1}{7}+2$$

Whakareatia -5 ki te \frac{1}{7}.

$$a=-\frac{5}{7}+2$$

Tāpiri 2 ki te -\frac{5}{7}.

$$a=\frac{9}{7}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{9}{7},b=\frac{1}{7}$$

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.

$$a+5b=2,a-2b=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&5\\1&-2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}2\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&5\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&5\\1&-2\end{matrix}\right))\left(\begin{matrix}1&5\\1&-2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&-2\end{matrix}\right))\left(\begin{matrix}2\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&-2\end{matrix}\right))\left(\begin{matrix}2\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&-2\end{matrix}\right))\left(\begin{matrix}2\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-5}&-\frac{5}{-2-5}\\-\frac{1}{-2-5}&\frac{1}{-2-5}\end{matrix}\right)\left(\begin{matrix}2\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{5}{7}\\\frac{1}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}2\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 2+\frac{5}{7}\\\frac{1}{7}\times 2-\frac{1}{7}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{9}{7}\\\frac{1}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{9}{7},b=\frac{1}{7}$$

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.

$$a+5b=2,a-2b=1$$

Me tango a-2b=1 mai i a+5b=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-a+5b+2b=2-1$$

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

$$5b+2b=2-1$$

Tāpiri 5b ki te 2b.

$$7b=2-1$$

Tāpiri 2 ki te -1.

$$7b=1$$

Whakawehea ngā taha e rua ki te 7.

$$b=\frac{1}{7}$$

Whakaurua te \frac{1}{7} mō b ki a-2b=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a-2\times \frac{1}{7}=1$$

Whakareatia -2 ki te \frac{1}{7}.

$$a-\frac{2}{7}=1$$

Me tāpiri \frac{2}{7} ki ngā taha e rua o te whārite.

$$a=\frac{9}{7}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{9}{7},b=\frac{1}{7}$$