Whakaoti i te {l}{5x+5y=14}{2x+4y=10}

$$\left. \begin{array} { l } { 5 x + 5 y = 14 } \\ { 2 x + 4 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{3}{5}=0.6$$

$$y = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

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.

$$5x+5y=14,2x+4y=10$$

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

$$5x+5y=14$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$5x=-5y+14$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-5y+14\right)$$

Whakareatia \frac{1}{5} ki te -5y+14.

$$x=-y+\frac{14}{5}$$

Whakakapia te -y+\frac{14}{5} mō te x ki tērā atu whārite, 2x+4y=10.

$$2\left(-y+\frac{14}{5}\right)+4y=10$$

Whakareatia 2 ki te -y+\frac{14}{5}.

$$-2y+\frac{28}{5}+4y=10$$

Tāpiri -2y ki te 4y.

$$2y+\frac{28}{5}=10$$

Me tango \frac{28}{5} mai i ngā taha e rua o te whārite.

$$2y=\frac{22}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} mō y ki x=-y+\frac{14}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{11}{5}+\frac{14}{5}$$

Whakareatia -1 ki te \frac{11}{5}.

$$x=\frac{-11+14}{5}$$

Tāpiri \frac{14}{5} ki te -\frac{11}{5} 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.

$$x=\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{5},y=\frac{11}{5}$$

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.

$$5x+5y=14,2x+4y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&5\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&5\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&5\\2&4\end{matrix}\right))\left(\begin{matrix}5&5\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&5\\2&4\end{matrix}\right))\left(\begin{matrix}14\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&5\\2&4\end{matrix}\right))\left(\begin{matrix}14\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&5\\2&4\end{matrix}\right))\left(\begin{matrix}14\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-5\times 2}&-\frac{5}{5\times 4-5\times 2}\\-\frac{2}{5\times 4-5\times 2}&\frac{5}{5\times 4-5\times 2}\end{matrix}\right)\left(\begin{matrix}14\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{2}\\-\frac{1}{5}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}14\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 14-\frac{1}{2}\times 10\\-\frac{1}{5}\times 14+\frac{1}{2}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\\\frac{11}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{5},y=\frac{11}{5}$$

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.

$$5x+5y=14,2x+4y=10$$

Kia ōrite ai a 5x 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 5.

$$2\times 5x+2\times 5y=2\times 14,5\times 2x+5\times 4y=5\times 10$$

Whakarūnātia.

$$10x+10y=28,10x+20y=50$$

Me tango 10x+20y=50 mai i 10x+10y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+10y-20y=28-50$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-20y=28-50$$

Tāpiri 10y ki te -20y.

$$-10y=28-50$$

Tāpiri 28 ki te -50.

$$-10y=-22$$

Whakawehea ngā taha e rua ki te -10.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} mō y ki 2x+4y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times \frac{11}{5}=10$$

Whakareatia 4 ki te \frac{11}{5}.

$$2x+\frac{44}{5}=10$$

Me tango \frac{44}{5} mai i ngā taha e rua o te whārite.

$$2x=\frac{6}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{5},y=\frac{11}{5}$$