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

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

Whakaoti mō x, y

$$x=-1$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=2$$

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 tāpiri y ki 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 tāpiri 18 ki 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

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=2$$

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.

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

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.

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

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

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=2$$

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.

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

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 tango 5 mai i 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.

$$x=-1,y=5$$