Whakaoti i te {l}{12x+4y=6}{9x+16y=8}

$$\left. \begin{array} { l } { 12 x + 4 y = 6 } \\ { 9 x + 16 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{16}{39}\approx 0.41025641$$

$$y=\frac{7}{26}\approx 0.269230769$$

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.

$$12x+4y=6,9x+16y=8$$

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

$$12x+4y=6$$

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

$$12x=-4y+6$$

Whakawehea ngā taha e rua ki te 12.

$$x=\frac{1}{12}\left(-4y+6\right)$$

Whakareatia \frac{1}{12} ki te -4y+6.

$$x=-\frac{1}{3}y+\frac{1}{2}$$

Whakakapia te -\frac{y}{3}+\frac{1}{2} mō te x ki tērā atu whārite, 9x+16y=8.

$$9\left(-\frac{1}{3}y+\frac{1}{2}\right)+16y=8$$

Whakareatia 9 ki te -\frac{y}{3}+\frac{1}{2}.

$$-3y+\frac{9}{2}+16y=8$$

Tāpiri -3y ki te 16y.

$$13y+\frac{9}{2}=8$$

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

$$13y=\frac{7}{2}$$

Whakawehea ngā taha e rua ki te 13.

$$y=\frac{7}{26}$$

Whakaurua te \frac{7}{26} mō y ki x=-\frac{1}{3}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{7}{26}+\frac{1}{2}$$

Whakareatia -\frac{1}{3} ki te \frac{7}{26} 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.

$$x=-\frac{7}{78}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te -\frac{7}{78} 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.

$$x=\frac{16}{39}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{39},y=\frac{7}{26}$$

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.

$$12x+4y=6,9x+16y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}12&4\\9&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}12&4\\9&16\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&4\\9&16\end{matrix}\right))\left(\begin{matrix}12&4\\9&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\9&16\end{matrix}\right))\left(\begin{matrix}6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\9&16\end{matrix}\right))\left(\begin{matrix}6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\9&16\end{matrix}\right))\left(\begin{matrix}6\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{12\times 16-4\times 9}&-\frac{4}{12\times 16-4\times 9}\\-\frac{9}{12\times 16-4\times 9}&\frac{12}{12\times 16-4\times 9}\end{matrix}\right)\left(\begin{matrix}6\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{39}&-\frac{1}{39}\\-\frac{3}{52}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}6\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{39}\times 6-\frac{1}{39}\times 8\\-\frac{3}{52}\times 6+\frac{1}{13}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{39}\\\frac{7}{26}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{39},y=\frac{7}{26}$$

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.

$$12x+4y=6,9x+16y=8$$

Kia ōrite ai a 12x me 9x, 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 12.

$$9\times 12x+9\times 4y=9\times 6,12\times 9x+12\times 16y=12\times 8$$

Whakarūnātia.

$$108x+36y=54,108x+192y=96$$

Me tango 108x+192y=96 mai i 108x+36y=54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$108x-108x+36y-192y=54-96$$

Tāpiri 108x ki te -108x. Ka whakakore atu ngā kupu 108x me -108x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36y-192y=54-96$$

Tāpiri 36y ki te -192y.

$$-156y=54-96$$

Tāpiri 54 ki te -96.

$$-156y=-42$$

Whakawehea ngā taha e rua ki te -156.

$$y=\frac{7}{26}$$

Whakaurua te \frac{7}{26} mō y ki 9x+16y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x+16\times \frac{7}{26}=8$$

Whakareatia 16 ki te \frac{7}{26}.

$$9x+\frac{56}{13}=8$$

Me tango \frac{56}{13} mai i ngā taha e rua o te whārite.

$$9x=\frac{48}{13}$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{16}{39}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{39},y=\frac{7}{26}$$