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

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

Whakaoti mō x, y

$$x=1$$

$$y=5$$

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.

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

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

$$x+y=6$$

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

$$x=-y+6$$

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

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

Whakareatia 3 ki te -y+6.

$$-3y+18-y=-2$$

Tāpiri -3y ki te -y.

$$-4y+18=-2$$

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

$$-4y=-20$$

Whakawehea ngā taha e rua ki te -4.

$$y=5$$

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

$$x=-5+6$$

Tāpiri 6 ki te -5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=5$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=5$$

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.

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

Kia ōrite ai a x 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 1.

$$3x+3y=3\times 6,3x-y=-2$$

Whakarūnātia.

$$3x+3y=18,3x-y=-2$$

Me tango 3x-y=-2 mai i 3x+3y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+3y+y=18+2$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+y=18+2$$

Tāpiri 3y ki te y.

$$4y=18+2$$

Tāpiri 18 ki te 2.

$$4y=20$$

Whakawehea ngā taha e rua ki te 4.

$$y=5$$

Whakaurua te 5 mō y ki 3x-y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-5=-2$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=5$$