Whakaoti i te {l}{4a+5v=28}{6a+3v=24}

$$\left. \begin{array} { l } { 4 a + 5 v = 28 } \\ { 6 a + 3 v = 24 } \end{array} \right.$$

Whakaoti mō a, v

$$a=2$$

$$v=4$$

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.

$$4a+5v=28,6a+3v=24$$

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

$$4a+5v=28$$

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

$$4a=-5v+28$$

Whakawehea ngā taha e rua ki te 4.

$$a=\frac{1}{4}\left(-5v+28\right)$$

Whakareatia \frac{1}{4} ki te -5v+28.

$$a=-\frac{5}{4}v+7$$

Whakakapia te -\frac{5v}{4}+7 mō te a ki tērā atu whārite, 6a+3v=24.

$$6\left(-\frac{5}{4}v+7\right)+3v=24$$

Whakareatia 6 ki te -\frac{5v}{4}+7.

$$-\frac{15}{2}v+42+3v=24$$

Tāpiri -\frac{15v}{2} ki te 3v.

$$-\frac{9}{2}v+42=24$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$-\frac{9}{2}v=-18$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$v=4$$

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

$$a=-\frac{5}{4}\times 4+7$$

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

$$a=-5+7$$

Tāpiri 7 ki te -5.

$$a=2$$

Kua oti te pūnaha te whakatau.

$$a=2,v=4$$

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.

$$4a+5v=28,6a+3v=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\6&3\end{matrix}\right)\left(\begin{matrix}a\\v\end{matrix}\right)=\left(\begin{matrix}28\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\6&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\6&3\end{matrix}\right))\left(\begin{matrix}4&5\\6&3\end{matrix}\right)\left(\begin{matrix}a\\v\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\6&3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\v\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\6&3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\v\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\6&3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\v\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-5\times 6}&-\frac{5}{4\times 3-5\times 6}\\-\frac{6}{4\times 3-5\times 6}&\frac{4}{4\times 3-5\times 6}\end{matrix}\right)\left(\begin{matrix}28\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\v\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{5}{18}\\\frac{1}{3}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}28\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\v\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 28+\frac{5}{18}\times 24\\\frac{1}{3}\times 28-\frac{2}{9}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\v\end{matrix}\right)=\left(\begin{matrix}2\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me v.

$$a=2,v=4$$

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.

$$4a+5v=28,6a+3v=24$$

Kia ōrite ai a 4a me 6a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$6\times 4a+6\times 5v=6\times 28,4\times 6a+4\times 3v=4\times 24$$

Whakarūnātia.

$$24a+30v=168,24a+12v=96$$

Me tango 24a+12v=96 mai i 24a+30v=168 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24a-24a+30v-12v=168-96$$

Tāpiri 24a ki te -24a. Ka whakakore atu ngā kupu 24a me -24a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30v-12v=168-96$$

Tāpiri 30v ki te -12v.

$$18v=168-96$$

Tāpiri 168 ki te -96.

$$18v=72$$

Whakawehea ngā taha e rua ki te 18.

$$v=4$$

Whakaurua te 4 mō v ki 6a+3v=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$6a+3\times 4=24$$

Whakareatia 3 ki te 4.

$$6a+12=24$$

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

$$6a=12$$

Whakawehea ngā taha e rua ki te 6.

$$a=2$$

Kua oti te pūnaha te whakatau.

$$a=2,v=4$$