Whakaoti i te {l}{6u+4v=5}{9u-8v=4}

$$\left\{ \begin{array} { l } { 6 u + 4 v = 5 } \\ { 9 u - 8 v = 4 } \end{array} \right.$$

Whakaoti mō u, v

$$u=\frac{2}{3}\approx 0.666666667$$

$$v=\frac{1}{4}=0.25$$

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.

$$6u+4v=5,9u-8v=4$$

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

$$6u+4v=5$$

Me tango 4v mai i ngā taha e rua o te whārite.

$$6u=-4v+5$$

Whakawehea ngā taha e rua ki te 6.

$$u=\frac{1}{6}\left(-4v+5\right)$$

Whakareatia \frac{1}{6} ki te -4v+5.

$$u=-\frac{2}{3}v+\frac{5}{6}$$

Whakakapia te -\frac{2v}{3}+\frac{5}{6} mō te u ki tērā atu whārite, 9u-8v=4.

$$9\left(-\frac{2}{3}v+\frac{5}{6}\right)-8v=4$$

Whakareatia 9 ki te -\frac{2v}{3}+\frac{5}{6}.

$$-6v+\frac{15}{2}-8v=4$$

Tāpiri -6v ki te -8v.

$$-14v+\frac{15}{2}=4$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$-14v=-\frac{7}{2}$$

Whakawehea ngā taha e rua ki te -14.

$$v=\frac{1}{4}$$

Whakaurua te \frac{1}{4} mō v ki u=-\frac{2}{3}v+\frac{5}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$u=-\frac{2}{3}\times \frac{1}{4}+\frac{5}{6}$$

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

$$u=\frac{-1+5}{6}$$

Tāpiri \frac{5}{6} ki te -\frac{1}{6} 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.

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

Kua oti te pūnaha te whakatau.

$$u=\frac{2}{3},v=\frac{1}{4}$$

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.

$$6u+4v=5,9u-8v=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&4\\9&-8\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}5\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&4\\9&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&4\\9&-8\end{matrix}\right))\left(\begin{matrix}6&4\\9&-8\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\9&-8\end{matrix}\right))\left(\begin{matrix}5\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\9&-8\end{matrix}\right))\left(\begin{matrix}5\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\9&-8\end{matrix}\right))\left(\begin{matrix}5\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{6\left(-8\right)-4\times 9}&-\frac{4}{6\left(-8\right)-4\times 9}\\-\frac{9}{6\left(-8\right)-4\times 9}&\frac{6}{6\left(-8\right)-4\times 9}\end{matrix}\right)\left(\begin{matrix}5\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}\frac{2}{21}&\frac{1}{21}\\\frac{3}{28}&-\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}5\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}\frac{2}{21}\times 5+\frac{1}{21}\times 4\\\frac{3}{28}\times 5-\frac{1}{14}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\\\frac{1}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa u me v.

$$u=\frac{2}{3},v=\frac{1}{4}$$

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.

$$6u+4v=5,9u-8v=4$$

Kia ōrite ai a 6u me 9u, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$9\times 6u+9\times 4v=9\times 5,6\times 9u+6\left(-8\right)v=6\times 4$$

Whakarūnātia.

$$54u+36v=45,54u-48v=24$$

Me tango 54u-48v=24 mai i 54u+36v=45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$54u-54u+36v+48v=45-24$$

Tāpiri 54u ki te -54u. Ka whakakore atu ngā kupu 54u me -54u, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36v+48v=45-24$$

Tāpiri 36v ki te 48v.

$$84v=45-24$$

Tāpiri 45 ki te -24.

$$84v=21$$

Whakawehea ngā taha e rua ki te 84.

$$v=\frac{1}{4}$$

Whakaurua te \frac{1}{4} mō v ki 9u-8v=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$9u-8\times \frac{1}{4}=4$$

Whakareatia -8 ki te \frac{1}{4}.

$$9u-2=4$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$9u=6$$

Whakawehea ngā taha e rua ki te 9.

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

Kua oti te pūnaha te whakatau.

$$u=\frac{2}{3},v=\frac{1}{4}$$