Whakaoti i te {l}{2x+2y=0}{3x-y=2}

$$\left. \begin{array} { l } { 2 x + 2 y = 0 } \\ { 3 x - y = 2 } \end{array} \right.$$

Whakaoti mō x, y

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

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

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$2x+2y=0,3x-y=2$$

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

$$2x+2y=0$$

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

$$2x=-2y$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-y$$

Whakakapia te -y mō te x ki tērā atu whārite, 3x-y=2.

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

Whakareatia 3 ki te -y.

$$-3y-y=2$$

Tāpiri -3y ki te -y.

$$-4y=2$$

Whakawehea ngā taha e rua ki te -4.

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

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

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

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

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

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$2x+2y=0,3x-y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&2\\3&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&2\\3&-1\end{matrix}\right))\left(\begin{matrix}2&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\3&-1\end{matrix}\right))\left(\begin{matrix}0\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\3&-1\end{matrix}\right))\left(\begin{matrix}0\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\3&-1\end{matrix}\right))\left(\begin{matrix}0\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-2\times 3}&-\frac{2}{2\left(-1\right)-2\times 3}\\-\frac{3}{2\left(-1\right)-2\times 3}&\frac{2}{2\left(-1\right)-2\times 3}\end{matrix}\right)\left(\begin{matrix}0\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&\frac{1}{4}\\\frac{3}{8}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}0\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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.

$$2x+2y=0,3x-y=2$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\times 2y=0,2\times 3x+2\left(-1\right)y=2\times 2$$

Whakarūnātia.

$$6x+6y=0,6x-2y=4$$

Me tango 6x-2y=4 mai i 6x+6y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri 6y ki te 2y.

$$8y=-4$$

Whakawehea ngā taha e rua ki te 8.

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

Whakaurua te -\frac{1}{2} 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.

$$3x-\left(-\frac{1}{2}\right)=2$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$3x=\frac{3}{2}$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

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