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

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

Whakaoti mō y, x

$$x=2$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+x=6$$

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.

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

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

$$y+x=6$$

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

$$y=-x+6$$

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

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

Whakareatia -2 ki te -x+6.

$$2x-12+x=-6$$

Tāpiri 2x ki te x.

$$3x-12=-6$$

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

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

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

$$y=-2+6$$

Tāpiri 6 ki te -2.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+x=6$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=4,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+x=6$$

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.

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

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

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

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

$$y+2y=6+6$$

Tāpiri y ki te 2y.

$$3y=6+6$$

Tāpiri 6 ki te 6.

$$3y=12$$

Whakawehea ngā taha e rua ki te 3.

$$y=4$$

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

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

Whakareatia -2 ki te 4.

$$-8+x=-6$$

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

$$x=2$$

Kua oti te pūnaha te whakatau.

$$y=4,x=2$$