Whakaoti i te {r}{8x+y=21}{24x-5y=23}

$$\left. \begin{array} { r } { 8 x + y = 21 } \\ { 24 x - 5 y = 23 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=5$$

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.

$$8x+y=21,24x-5y=23$$

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

$$8x+y=21$$

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

$$8x=-y+21$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-y+21\right)$$

Whakareatia \frac{1}{8} ki te -y+21.

$$x=-\frac{1}{8}y+\frac{21}{8}$$

Whakakapia te \frac{-y+21}{8} mō te x ki tērā atu whārite, 24x-5y=23.

$$24\left(-\frac{1}{8}y+\frac{21}{8}\right)-5y=23$$

Whakareatia 24 ki te \frac{-y+21}{8}.

$$-3y+63-5y=23$$

Tāpiri -3y ki te -5y.

$$-8y+63=23$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$-8y=-40$$

Whakawehea ngā taha e rua ki te -8.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{1}{8}y+\frac{21}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{8}\times 5+\frac{21}{8}$$

Whakareatia -\frac{1}{8} ki te 5.

$$x=\frac{-5+21}{8}$$

Tāpiri \frac{21}{8} ki te -\frac{5}{8} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$

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.

$$8x+y=21,24x-5y=23$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&1\\24&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}21\\23\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&1\\24&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&1\\24&-5\end{matrix}\right))\left(\begin{matrix}8&1\\24&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\24&-5\end{matrix}\right))\left(\begin{matrix}21\\23\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\24&-5\end{matrix}\right))\left(\begin{matrix}21\\23\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\24&-5\end{matrix}\right))\left(\begin{matrix}21\\23\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{8\left(-5\right)-24}&-\frac{1}{8\left(-5\right)-24}\\-\frac{24}{8\left(-5\right)-24}&\frac{8}{8\left(-5\right)-24}\end{matrix}\right)\left(\begin{matrix}21\\23\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{64}&\frac{1}{64}\\\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}21\\23\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{64}\times 21+\frac{1}{64}\times 23\\\frac{3}{8}\times 21-\frac{1}{8}\times 23\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=5$$

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.

$$8x+y=21,24x-5y=23$$

Kia ōrite ai a 8x me 24x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 24 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$24\times 8x+24y=24\times 21,8\times 24x+8\left(-5\right)y=8\times 23$$

Whakarūnātia.

$$192x+24y=504,192x-40y=184$$

Me tango 192x-40y=184 mai i 192x+24y=504 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$192x-192x+24y+40y=504-184$$

Tāpiri 192x ki te -192x. Ka whakakore atu ngā kupu 192x me -192x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y+40y=504-184$$

Tāpiri 24y ki te 40y.

$$64y=504-184$$

Tāpiri 504 ki te -184.

$$64y=320$$

Whakawehea ngā taha e rua ki te 64.

$$y=5$$

Whakaurua te 5 mō y ki 24x-5y=23. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$24x-5\times 5=23$$

Whakareatia -5 ki te 5.

$$24x-25=23$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$24x=48$$

Whakawehea ngā taha e rua ki te 24.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$