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

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

Whakaoti mō x, y

$$x=2$$

$$y=0$$

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.

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

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

$$3x-y=6$$

Me tāpiri y ki ngā taha e rua o te whārite.

$$3x=y+6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+6\right)$$

Whakareatia \frac{1}{3} ki te y+6.

$$x=\frac{1}{3}y+2$$

Whakakapia te \frac{y}{3}+2 mō te x ki tērā atu whārite, 2x+y=4.

$$2\left(\frac{1}{3}y+2\right)+y=4$$

Whakareatia 2 ki te \frac{y}{3}+2.

$$\frac{2}{3}y+4+y=4$$

Tāpiri \frac{2y}{3} ki te y.

$$\frac{5}{3}y+4=4$$

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

$$\frac{5}{3}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{1}{3}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{5}\\-\frac{2}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 6+\frac{1}{5}\times 4\\-\frac{2}{5}\times 6+\frac{3}{5}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=0$$

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.

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

Kia ōrite ai a 3x 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 3.

$$2\times 3x+2\left(-1\right)y=2\times 6,3\times 2x+3y=3\times 4$$

Whakarūnātia.

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

Me tango 6x+3y=12 mai i 6x-2y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-2y-3y=12-12$$

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.

$$-2y-3y=12-12$$

Tāpiri -2y ki te -3y.

$$-5y=12-12$$

Tāpiri 12 ki te -12.

$$-5y=0$$

Whakawehea ngā taha e rua ki te -5.

$$y=0$$

Whakaurua te 0 mō y ki 2x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=0$$