Whakaoti i te {c}{x+y=5.90}{2x+4y=16.80}

$$\left. \begin{array} { c } { x + y = 5.90 } \\ { 2 x + 4 y = 16.80 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3.4$$

$$y=2.5$$

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.

$$x+y=5.9,2x+4y=16.8$$

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

$$x+y=5.9$$

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

$$x=-y+5.9$$

Whakakapia te -y+5.9 mō te x ki tērā atu whārite, 2x+4y=16.8.

$$2\left(-y+5.9\right)+4y=16.8$$

Whakareatia 2 ki te -y+5.9.

$$-2y+11.8+4y=16.8$$

Tāpiri -2y ki te 4y.

$$2y+11.8=16.8$$

Me tango 11.8 mai i ngā taha e rua o te whārite.

$$2y=5$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō y ki x=-y+5.9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}+5.9$$

Tāpiri 5.9 ki te -\frac{5}{2} 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.

$$x=\frac{17}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{17}{5},y=\frac{5}{2}$$

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.

$$x+y=5.9,2x+4y=16.8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5.9\\16.8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}1&1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}5.9\\16.8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}5.9\\16.8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}5.9\\16.8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-2}&-\frac{1}{4-2}\\-\frac{2}{4-2}&\frac{1}{4-2}\end{matrix}\right)\left(\begin{matrix}5.9\\16.8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-\frac{1}{2}\\-1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}5.9\\16.8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 5.9-\frac{1}{2}\times 16.8\\-5.9+\frac{1}{2}\times 16.8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{5}\\\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{17}{5},y=\frac{5}{2}$$

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.

$$x+y=5.9,2x+4y=16.8$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2y=2\times 5.9,2x+4y=16.8$$

Whakarūnātia.

$$2x+2y=11.8,2x+4y=16.8$$

Me tango 2x+4y=16.8 mai i 2x+2y=11.8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-4y=\frac{59-84}{5}$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-4y=\frac{59-84}{5}$$

Tāpiri 2y ki te -4y.

$$-2y=\frac{59-84}{5}$$

Tāpiri 11.8 ki te -16.8 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.

$$-2y=-5$$

Whakawehea ngā taha e rua ki te -2.

$$y=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō y ki 2x+4y=16.8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times \frac{5}{2}=16.8$$

Whakareatia 4 ki te \frac{5}{2}.

$$2x+10=16.8$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$2x=6.8$$

Whakawehea ngā taha e rua ki te 2.

$$x=3.4$$

Kua oti te pūnaha te whakatau.

$$x=3.4,y=\frac{5}{2}$$