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. Tangohia te 3x mai i 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 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.

$$x=-\left(-5\right)-6$$

Whakareatia -1 ki te -5.

$$x=5-6$$

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. Tangohia te 3x mai i 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.

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

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

Whakaarohia te whārite tuarua. Tangohia te 3x mai i 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.

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

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

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

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

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

Tāpiri x ki te 3x.

$$4x=-6+2$$

Tāpiri -6 ki te 2.

$$4x=-4$$

Whakawehea ngā taha e rua ki te 4.

$$x=-1$$

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

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

Whakareatia -3 ki te -1.

$$3+y=-2$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$y=-5$$

Kua oti te pūnaha te whakatau.

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