Whakaoti i te {l}{8x+y=67}{4x+7y=51}

$$\left. \begin{array} { l } { 8 x + y = 67 } \\ { 4 x + 7 y = 51 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{209}{26} = 8\frac{1}{26} \approx 8.038461538$$

$$y = \frac{35}{13} = 2\frac{9}{13} \approx 2.692307692$$

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.

$$8x+y=67,4x+7y=51$$

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

$$8x+y=67$$

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

$$8x=-y+67$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-y+67\right)$$

Whakareatia \frac{1}{8} ki te -y+67.

$$x=-\frac{1}{8}y+\frac{67}{8}$$

Whakakapia te \frac{-y+67}{8} mō te x ki tērā atu whārite, 4x+7y=51.

$$4\left(-\frac{1}{8}y+\frac{67}{8}\right)+7y=51$$

Whakareatia 4 ki te \frac{-y+67}{8}.

$$-\frac{1}{2}y+\frac{67}{2}+7y=51$$

Tāpiri -\frac{y}{2} ki te 7y.

$$\frac{13}{2}y+\frac{67}{2}=51$$

Me tango \frac{67}{2} mai i ngā taha e rua o te whārite.

$$\frac{13}{2}y=\frac{35}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{35}{13}$$

Whakaurua te \frac{35}{13} mō y ki x=-\frac{1}{8}y+\frac{67}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{8}\times \frac{35}{13}+\frac{67}{8}$$

Whakareatia -\frac{1}{8} ki te \frac{35}{13} 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.

$$x=-\frac{35}{104}+\frac{67}{8}$$

Tāpiri \frac{67}{8} ki te -\frac{35}{104} 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.

$$x=\frac{209}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{209}{26},y=\frac{35}{13}$$

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.

$$8x+y=67,4x+7y=51$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&1\\4&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}67\\51\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&1\\4&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&1\\4&7\end{matrix}\right))\left(\begin{matrix}8&1\\4&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\4&7\end{matrix}\right))\left(\begin{matrix}67\\51\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\4&7\end{matrix}\right))\left(\begin{matrix}67\\51\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\4&7\end{matrix}\right))\left(\begin{matrix}67\\51\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{8\times 7-4}&-\frac{1}{8\times 7-4}\\-\frac{4}{8\times 7-4}&\frac{8}{8\times 7-4}\end{matrix}\right)\left(\begin{matrix}67\\51\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{52}&-\frac{1}{52}\\-\frac{1}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}67\\51\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{52}\times 67-\frac{1}{52}\times 51\\-\frac{1}{13}\times 67+\frac{2}{13}\times 51\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{209}{26}\\\frac{35}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{209}{26},y=\frac{35}{13}$$

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.

$$8x+y=67,4x+7y=51$$

Kia ōrite ai a 8x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$4\times 8x+4y=4\times 67,8\times 4x+8\times 7y=8\times 51$$

Whakarūnātia.

$$32x+4y=268,32x+56y=408$$

Me tango 32x+56y=408 mai i 32x+4y=268 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$32x-32x+4y-56y=268-408$$

Tāpiri 32x ki te -32x. Ka whakakore atu ngā kupu 32x me -32x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-56y=268-408$$

Tāpiri 4y ki te -56y.

$$-52y=268-408$$

Tāpiri 268 ki te -408.

$$-52y=-140$$

Whakawehea ngā taha e rua ki te -52.

$$y=\frac{35}{13}$$

Whakaurua te \frac{35}{13} mō y ki 4x+7y=51. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+7\times \frac{35}{13}=51$$

Whakareatia 7 ki te \frac{35}{13}.

$$4x+\frac{245}{13}=51$$

Me tango \frac{245}{13} mai i ngā taha e rua o te whārite.

$$4x=\frac{418}{13}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{209}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{209}{26},y=\frac{35}{13}$$