Whakaoti i te {l}{9x+2y=62}{4x+4y=36}

$$\left\{ \begin{array} { l } { 9 x + 2 y = 62 } \\ { 4 x + 4 y = 36 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{44}{7} = 6\frac{2}{7} \approx 6.285714286$$

$$y = \frac{19}{7} = 2\frac{5}{7} \approx 2.714285714$$

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.

$$9x+2y=62,4x+4y=36$$

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

$$9x+2y=62$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$9x=-2y+62$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(-2y+62\right)$$

Whakareatia \frac{1}{9} ki te -2y+62.

$$x=-\frac{2}{9}y+\frac{62}{9}$$

Whakakapia te \frac{-2y+62}{9} mō te x ki tērā atu whārite, 4x+4y=36.

$$4\left(-\frac{2}{9}y+\frac{62}{9}\right)+4y=36$$

Whakareatia 4 ki te \frac{-2y+62}{9}.

$$-\frac{8}{9}y+\frac{248}{9}+4y=36$$

Tāpiri -\frac{8y}{9} ki te 4y.

$$\frac{28}{9}y+\frac{248}{9}=36$$

Me tango \frac{248}{9} mai i ngā taha e rua o te whārite.

$$\frac{28}{9}y=\frac{76}{9}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{28}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{19}{7}$$

Whakaurua te \frac{19}{7} mō y ki x=-\frac{2}{9}y+\frac{62}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{9}\times \frac{19}{7}+\frac{62}{9}$$

Whakareatia -\frac{2}{9} ki te \frac{19}{7} 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.

$$x=-\frac{38}{63}+\frac{62}{9}$$

Tāpiri \frac{62}{9} ki te -\frac{38}{63} 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.

$$x=\frac{44}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{44}{7},y=\frac{19}{7}$$

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.

$$9x+2y=62,4x+4y=36$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&2\\4&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}62\\36\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&2\\4&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&2\\4&4\end{matrix}\right))\left(\begin{matrix}9&2\\4&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&2\\4&4\end{matrix}\right))\left(\begin{matrix}62\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&2\\4&4\end{matrix}\right))\left(\begin{matrix}62\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&2\\4&4\end{matrix}\right))\left(\begin{matrix}62\\36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9\times 4-2\times 4}&-\frac{2}{9\times 4-2\times 4}\\-\frac{4}{9\times 4-2\times 4}&\frac{9}{9\times 4-2\times 4}\end{matrix}\right)\left(\begin{matrix}62\\36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{1}{14}\\-\frac{1}{7}&\frac{9}{28}\end{matrix}\right)\left(\begin{matrix}62\\36\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 62-\frac{1}{14}\times 36\\-\frac{1}{7}\times 62+\frac{9}{28}\times 36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{44}{7}\\\frac{19}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{44}{7},y=\frac{19}{7}$$

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.

$$9x+2y=62,4x+4y=36$$

Kia ōrite ai a 9x 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 9.

$$4\times 9x+4\times 2y=4\times 62,9\times 4x+9\times 4y=9\times 36$$

Whakarūnātia.

$$36x+8y=248,36x+36y=324$$

Me tango 36x+36y=324 mai i 36x+8y=248 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$36x-36x+8y-36y=248-324$$

Tāpiri 36x ki te -36x. Ka whakakore atu ngā kupu 36x me -36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-36y=248-324$$

Tāpiri 8y ki te -36y.

$$-28y=248-324$$

Tāpiri 248 ki te -324.

$$-28y=-76$$

Whakawehea ngā taha e rua ki te -28.

$$y=\frac{19}{7}$$

Whakaurua te \frac{19}{7} mō y ki 4x+4y=36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+4\times \frac{19}{7}=36$$

Whakareatia 4 ki te \frac{19}{7}.

$$4x+\frac{76}{7}=36$$

Me tango \frac{76}{7} mai i ngā taha e rua o te whārite.

$$4x=\frac{176}{7}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{44}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{44}{7},y=\frac{19}{7}$$