Whakaoti i te {l}{x+y=18}{4x+2y=52}

$$\left. \begin{array} { l } { x + y = 18 } \\ { 4 x + 2 y = 52 } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=10$$

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.

$$x+y=18,4x+2y=52$$

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

$$x+y=18$$

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

$$x=-y+18$$

Whakakapia te -y+18 mō te x ki tērā atu whārite, 4x+2y=52.

$$4\left(-y+18\right)+2y=52$$

Whakareatia 4 ki te -y+18.

$$-4y+72+2y=52$$

Tāpiri -4y ki te 2y.

$$-2y+72=52$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-2y=-20$$

Whakawehea ngā taha e rua ki te -2.

$$y=10$$

Whakaurua te 10 mō y ki x=-y+18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-10+18$$

Tāpiri 18 ki te -10.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=10$$

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.

$$x+y=18,4x+2y=52$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\52\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\4&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}18\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}18\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}18\\52\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-4}&-\frac{1}{2-4}\\-\frac{4}{2-4}&\frac{1}{2-4}\end{matrix}\right)\left(\begin{matrix}18\\52\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&\frac{1}{2}\\2&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}18\\52\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18+\frac{1}{2}\times 52\\2\times 18-\frac{1}{2}\times 52\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=10$$

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.

$$x+y=18,4x+2y=52$$

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.

$$4x+4y=4\times 18,4x+2y=52$$

Whakarūnātia.

$$4x+4y=72,4x+2y=52$$

Me tango 4x+2y=52 mai i 4x+4y=72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+4y-2y=72-52$$

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.

$$4y-2y=72-52$$

Tāpiri 4y ki te -2y.

$$2y=72-52$$

Tāpiri 72 ki te -52.

$$2y=20$$

Whakawehea ngā taha e rua ki te 2.

$$y=10$$

Whakaurua te 10 mō y ki 4x+2y=52. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+2\times 10=52$$

Whakareatia 2 ki te 10.

$$4x+20=52$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$4x=32$$

Whakawehea ngā taha e rua ki te 4.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=10$$