Whakaoti i te {c}{y=2-2x}{5y+2x=14}

$$\left\{ \begin{array} { c } { y = 2 - 2 x } \\ { 5 y + 2 x = 14 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-\frac{1}{2}=-0.5$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=2$$

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.

$$y+2x=2,5y+2x=14$$

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

$$y+2x=2$$

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

$$y=-2x+2$$

Whakakapia te -2x+2 mō te y ki tērā atu whārite, 5y+2x=14.

$$5\left(-2x+2\right)+2x=14$$

Whakareatia 5 ki te -2x+2.

$$-10x+10+2x=14$$

Tāpiri -10x ki te 2x.

$$-8x+10=14$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-8x=4$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{2}$$

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

$$y=-2\left(-\frac{1}{2}\right)+2$$

Whakareatia -2 ki te -\frac{1}{2}.

$$y=1+2$$

Tāpiri 2 ki te 1.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=-\frac{1}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=2$$

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

$$y+2x=2,5y+2x=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\5&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\5&2\end{matrix}\right))\left(\begin{matrix}1&2\\5&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&2\end{matrix}\right))\left(\begin{matrix}2\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&2\end{matrix}\right))\left(\begin{matrix}2\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&2\end{matrix}\right))\left(\begin{matrix}2\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-2\times 5}&-\frac{2}{2-2\times 5}\\-\frac{5}{2-2\times 5}&\frac{1}{2-2\times 5}\end{matrix}\right)\left(\begin{matrix}2\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{5}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}2\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 2+\frac{1}{4}\times 14\\\frac{5}{8}\times 2-\frac{1}{8}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=3,x=-\frac{1}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=2$$

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.

$$y+2x=2,5y+2x=14$$

Me tango 5y+2x=14 mai i y+2x=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-5y+2x-2x=2-14$$

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

$$y-5y=2-14$$

Tāpiri y ki te -5y.

$$-4y=2-14$$

Tāpiri 2 ki te -14.

$$-4y=-12$$

Whakawehea ngā taha e rua ki te -4.

$$y=3$$

Whakaurua te 3 mō y ki 5y+2x=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5\times 3+2x=14$$

Whakareatia 5 ki te 3.

$$15+2x=14$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$2x=-1$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$y=3,x=-\frac{1}{2}$$