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

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

Whakaoti mō x, y

$$x=4$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$2y-2x=-6$$

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.

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

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

$$x+2y=6$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y+6$$

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

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

Whakareatia -2 ki te -2y+6.

$$4y-12+2y=-6$$

Tāpiri 4y ki te 2y.

$$6y-12=-6$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$6y=6$$

Whakawehea ngā taha e rua ki te 6.

$$y=1$$

Whakaurua te 1 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.

$$x=-2+6$$

Tāpiri 6 ki te -2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$2y-2x=-6$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$2y-2x=-6$$

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.

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

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

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

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$x+2x=6+6$$

Tāpiri x ki te 2x.

$$3x=6+6$$

Tāpiri 6 ki te 6.

$$3x=12$$

Whakawehea ngā taha e rua ki te 3.

$$x=4$$

Whakaurua te 4 mō x ki -2x+2y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2\times 4+2y=-6$$

Whakareatia -2 ki te 4.

$$-8+2y=-6$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$2y=2$$

Whakawehea ngā taha e rua ki te 2.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$x=4,y=1$$