Whakaoti i te {c}{x+y=9}{4x+5y=39}

$$\left. \begin{array} { c } { x + y = 9 } \\ { 4 x + 5 y = 39 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=3$$

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.

$$x+y=9,4x+5y=39$$

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

$$x+y=9$$

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

$$x=-y+9$$

Whakakapia te -y+9 mō te x ki tērā atu whārite, 4x+5y=39.

$$4\left(-y+9\right)+5y=39$$

Whakareatia 4 ki te -y+9.

$$-4y+36+5y=39$$

Tāpiri -4y ki te 5y.

$$y+36=39$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$y=3$$

Whakaurua te 3 mō y ki x=-y+9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3+9$$

Tāpiri 9 ki te -3.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=3$$

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.

$$x+y=9,4x+5y=39$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\39\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\4&5\end{matrix}\right))\left(\begin{matrix}1&1\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&5\end{matrix}\right))\left(\begin{matrix}9\\39\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&5\end{matrix}\right))\left(\begin{matrix}9\\39\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&5\end{matrix}\right))\left(\begin{matrix}9\\39\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-4}&-\frac{1}{5-4}\\-\frac{4}{5-4}&\frac{1}{5-4}\end{matrix}\right)\left(\begin{matrix}9\\39\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&-1\\-4&1\end{matrix}\right)\left(\begin{matrix}9\\39\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 9-39\\-4\times 9+39\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=3$$

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.

$$x+y=9,4x+5y=39$$

Kia ōrite ai a x 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 1.

$$4x+4y=4\times 9,4x+5y=39$$

Whakarūnātia.

$$4x+4y=36,4x+5y=39$$

Me tango 4x+5y=39 mai i 4x+4y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+4y-5y=36-39$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-5y=36-39$$

Tāpiri 4y ki te -5y.

$$-y=36-39$$

Tāpiri 36 ki te -39.

$$-y=-3$$

Whakawehea ngā taha e rua ki te -1.

$$y=3$$

Whakaurua te 3 mō y ki 4x+5y=39. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+5\times 3=39$$

Whakareatia 5 ki te 3.

$$4x+15=39$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$4x=24$$

Whakawehea ngā taha e rua ki te 4.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=3$$