Whakaoti i te {l}{4a+5b=9}{2a-b=7}

$$\left. \begin{array} { l } { 4 a + 5 b = 9 } \\ { 2 a - b = 7 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{22}{7} = 3\frac{1}{7} \approx 3.142857143$$

$$b=-\frac{5}{7}\approx -0.714285714$$

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.

$$4a+5b=9,2a-b=7$$

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

$$4a+5b=9$$

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

$$4a=-5b+9$$

Whakawehea ngā taha e rua ki te 4.

$$a=\frac{1}{4}\left(-5b+9\right)$$

Whakareatia \frac{1}{4} ki te -5b+9.

$$a=-\frac{5}{4}b+\frac{9}{4}$$

Whakakapia te \frac{-5b+9}{4} mō te a ki tērā atu whārite, 2a-b=7.

$$2\left(-\frac{5}{4}b+\frac{9}{4}\right)-b=7$$

Whakareatia 2 ki te \frac{-5b+9}{4}.

$$-\frac{5}{2}b+\frac{9}{2}-b=7$$

Tāpiri -\frac{5b}{2} ki te -b.

$$-\frac{7}{2}b+\frac{9}{2}=7$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$-\frac{7}{2}b=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-\frac{5}{7}$$

Whakaurua te -\frac{5}{7} mō b ki a=-\frac{5}{4}b+\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{5}{4}\left(-\frac{5}{7}\right)+\frac{9}{4}$$

Whakareatia -\frac{5}{4} ki te -\frac{5}{7} 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.

$$a=\frac{25}{28}+\frac{9}{4}$$

Tāpiri \frac{9}{4} ki te \frac{25}{28} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=\frac{22}{7}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{22}{7},b=-\frac{5}{7}$$

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.

$$4a+5b=9,2a-b=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\2&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\2&-1\end{matrix}\right))\left(\begin{matrix}4&5\\2&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\2&-1\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\2&-1\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\2&-1\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-5\times 2}&-\frac{5}{4\left(-1\right)-5\times 2}\\-\frac{2}{4\left(-1\right)-5\times 2}&\frac{4}{4\left(-1\right)-5\times 2}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&\frac{5}{14}\\\frac{1}{7}&-\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\times 9+\frac{5}{14}\times 7\\\frac{1}{7}\times 9-\frac{2}{7}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{22}{7}\\-\frac{5}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{22}{7},b=-\frac{5}{7}$$

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.

$$4a+5b=9,2a-b=7$$

Kia ōrite ai a 4a me 2a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$2\times 4a+2\times 5b=2\times 9,4\times 2a+4\left(-1\right)b=4\times 7$$

Whakarūnātia.

$$8a+10b=18,8a-4b=28$$

Me tango 8a-4b=28 mai i 8a+10b=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8a-8a+10b+4b=18-28$$

Tāpiri 8a ki te -8a. Ka whakakore atu ngā kupu 8a me -8a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10b+4b=18-28$$

Tāpiri 10b ki te 4b.

$$14b=18-28$$

Tāpiri 18 ki te -28.

$$14b=-10$$

Whakawehea ngā taha e rua ki te 14.

$$b=-\frac{5}{7}$$

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

$$2a-\left(-\frac{5}{7}\right)=7$$

Me tango \frac{5}{7} mai i ngā taha e rua o te whārite.

$$2a=\frac{44}{7}$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{22}{7}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{22}{7},b=-\frac{5}{7}$$