Whakaoti i te {l}{2a+3b=4}{3a-8b=5}

$$\left\{ \begin{array} { l } { 2 a + 3 b = 4 } \\ { 3 a - 8 b = 5 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{47}{25} = 1\frac{22}{25} = 1.88$$

$$b=\frac{2}{25}=0.08$$

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.

$$2a+3b=4,3a-8b=5$$

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

$$2a+3b=4$$

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

$$2a=-3b+4$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{1}{2}\left(-3b+4\right)$$

Whakareatia \frac{1}{2} ki te -3b+4.

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

Whakakapia te -\frac{3b}{2}+2 mō te a ki tērā atu whārite, 3a-8b=5.

$$3\left(-\frac{3}{2}b+2\right)-8b=5$$

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

$$-\frac{9}{2}b+6-8b=5$$

Tāpiri -\frac{9b}{2} ki te -8b.

$$-\frac{25}{2}b+6=5$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-\frac{25}{2}b=-1$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{25}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=\frac{2}{25}$$

Whakaurua te \frac{2}{25} mō b ki a=-\frac{3}{2}b+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{3}{2}\times \frac{2}{25}+2$$

Whakareatia -\frac{3}{2} ki te \frac{2}{25} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=-\frac{3}{25}+2$$

Tāpiri 2 ki te -\frac{3}{25}.

$$a=\frac{47}{25}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{47}{25},b=\frac{2}{25}$$

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.

$$2a+3b=4,3a-8b=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&-8\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}4\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\3&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\3&-8\end{matrix}\right))\left(\begin{matrix}2&3\\3&-8\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&-8\end{matrix}\right))\left(\begin{matrix}4\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&-8\end{matrix}\right))\left(\begin{matrix}4\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&-8\end{matrix}\right))\left(\begin{matrix}4\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{2\left(-8\right)-3\times 3}&-\frac{3}{2\left(-8\right)-3\times 3}\\-\frac{3}{2\left(-8\right)-3\times 3}&\frac{2}{2\left(-8\right)-3\times 3}\end{matrix}\right)\left(\begin{matrix}4\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{8}{25}&\frac{3}{25}\\\frac{3}{25}&-\frac{2}{25}\end{matrix}\right)\left(\begin{matrix}4\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{8}{25}\times 4+\frac{3}{25}\times 5\\\frac{3}{25}\times 4-\frac{2}{25}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{47}{25}\\\frac{2}{25}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{47}{25},b=\frac{2}{25}$$

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.

$$2a+3b=4,3a-8b=5$$

Kia ōrite ai a 2a 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 2.

$$3\times 2a+3\times 3b=3\times 4,2\times 3a+2\left(-8\right)b=2\times 5$$

Whakarūnātia.

$$6a+9b=12,6a-16b=10$$

Me tango 6a-16b=10 mai i 6a+9b=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6a-6a+9b+16b=12-10$$

Tāpiri 6a ki te -6a. Ka whakakore atu ngā kupu 6a me -6a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9b+16b=12-10$$

Tāpiri 9b ki te 16b.

$$25b=12-10$$

Tāpiri 12 ki te -10.

$$25b=2$$

Whakawehea ngā taha e rua ki te 25.

$$b=\frac{2}{25}$$

Whakaurua te \frac{2}{25} mō b ki 3a-8b=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$3a-8\times \frac{2}{25}=5$$

Whakareatia -8 ki te \frac{2}{25}.

$$3a-\frac{16}{25}=5$$

Me tāpiri \frac{16}{25} ki ngā taha e rua o te whārite.

$$3a=\frac{141}{25}$$

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{47}{25}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{47}{25},b=\frac{2}{25}$$