Whakaoti i te {c}{x+y=55}{4x+2y=170}

$$\left. \begin{array} { c } { x + y = 55 } \\ { 4 x + 2 y = 170 } \end{array} \right.$$

Whakaoti mō x, y

$$x=30$$

$$y=25$$

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.

$$x+y=55,4x+2y=170$$

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

$$x+y=55$$

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

$$x=-y+55$$

Whakakapia te -y+55 mō te x ki tērā atu whārite, 4x+2y=170.

$$4\left(-y+55\right)+2y=170$$

Whakareatia 4 ki te -y+55.

$$-4y+220+2y=170$$

Tāpiri -4y ki te 2y.

$$-2y+220=170$$

Me tango 220 mai i ngā taha e rua o te whārite.

$$-2y=-50$$

Whakawehea ngā taha e rua ki te -2.

$$y=25$$

Whakaurua te 25 mō y ki x=-y+55. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-25+55$$

Tāpiri 55 ki te -25.

$$x=30$$

Kua oti te pūnaha te whakatau.

$$x=30,y=25$$

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.

$$x+y=55,4x+2y=170$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}55\\170\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}55\\170\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}55\\170\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}55\\170\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-4}&-\frac{1}{2-4}\\-\frac{4}{2-4}&\frac{1}{2-4}\end{matrix}\right)\left(\begin{matrix}55\\170\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&\frac{1}{2}\\2&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}55\\170\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-55+\frac{1}{2}\times 170\\2\times 55-\frac{1}{2}\times 170\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\25\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=30,y=25$$

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.

$$x+y=55,4x+2y=170$$

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.

$$4x+4y=4\times 55,4x+2y=170$$

Whakarūnātia.

$$4x+4y=220,4x+2y=170$$

Me tango 4x+2y=170 mai i 4x+4y=220 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+4y-2y=220-170$$

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.

$$4y-2y=220-170$$

Tāpiri 4y ki te -2y.

$$2y=220-170$$

Tāpiri 220 ki te -170.

$$2y=50$$

Whakawehea ngā taha e rua ki te 2.

$$y=25$$

Whakaurua te 25 mō y ki 4x+2y=170. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+2\times 25=170$$

Whakareatia 2 ki te 25.

$$4x+50=170$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$4x=120$$

Whakawehea ngā taha e rua ki te 4.

$$x=30$$

Kua oti te pūnaha te whakatau.

$$x=30,y=25$$