Whakaoti i te {l}{2x+4y=12}{3x+y=6}

$$\left. \begin{array} { l } { 2 x + 4 y = 12 } \\ { 3 x + y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

$$y = \frac{12}{5} = 2\frac{2}{5} = 2.4$$

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.

$$2x+4y=12,3x+y=6$$

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

$$2x+4y=12$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$2x=-4y+12$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-4y+12\right)$$

Whakareatia \frac{1}{2} ki te -4y+12.

$$x=-2y+6$$

Whakakapia te -2y+6 mō te x ki tērā atu whārite, 3x+y=6.

$$3\left(-2y+6\right)+y=6$$

Whakareatia 3 ki te -2y+6.

$$-6y+18+y=6$$

Tāpiri -6y ki te y.

$$-5y+18=6$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-5y=-12$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{12}{5}$$

Whakaurua te \frac{12}{5} mō y ki x=-2y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{12}{5}+6$$

Whakareatia -2 ki te \frac{12}{5}.

$$x=-\frac{24}{5}+6$$

Tāpiri 6 ki te -\frac{24}{5}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=\frac{12}{5}$$

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.

$$2x+4y=12,3x+y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&4\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&4\\3&1\end{matrix}\right))\left(\begin{matrix}2&4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\3&1\end{matrix}\right))\left(\begin{matrix}12\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\3&1\end{matrix}\right))\left(\begin{matrix}12\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\3&1\end{matrix}\right))\left(\begin{matrix}12\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-4\times 3}&-\frac{4}{2-4\times 3}\\-\frac{3}{2-4\times 3}&\frac{2}{2-4\times 3}\end{matrix}\right)\left(\begin{matrix}12\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}&\frac{2}{5}\\\frac{3}{10}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}12\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}\times 12+\frac{2}{5}\times 6\\\frac{3}{10}\times 12-\frac{1}{5}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\\\frac{12}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{6}{5},y=\frac{12}{5}$$

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.

$$2x+4y=12,3x+y=6$$

Kia ōrite ai a 2x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$3\times 2x+3\times 4y=3\times 12,2\times 3x+2y=2\times 6$$

Whakarūnātia.

$$6x+12y=36,6x+2y=12$$

Me tango 6x+2y=12 mai i 6x+12y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+12y-2y=36-12$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-2y=36-12$$

Tāpiri 12y ki te -2y.

$$10y=36-12$$

Tāpiri 36 ki te -12.

$$10y=24$$

Whakawehea ngā taha e rua ki te 10.

$$y=\frac{12}{5}$$

Whakaurua te \frac{12}{5} mō y ki 3x+y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+\frac{12}{5}=6$$

Me tango \frac{12}{5} mai i ngā taha e rua o te whārite.

$$3x=\frac{18}{5}$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=\frac{12}{5}$$