Whakaoti i te {c}{2x+y=3}{-2x-4y=-1}

$$\left. \begin{array} { c } { 2 x + y = 3 } \\ { - 2 x - 4 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{11}{6} = 1\frac{5}{6} \approx 1.833333333$$

$$y=-\frac{2}{3}\approx -0.666666667$$

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.

$$2x+y=3,-2x-4y=-1$$

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

$$2x+y=3$$

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

$$2x=-y+3$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -y+3.

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

Whakakapia te \frac{-y+3}{2} mō te x ki tērā atu whārite, -2x-4y=-1.

$$-2\left(-\frac{1}{2}y+\frac{3}{2}\right)-4y=-1$$

Whakareatia -2 ki te \frac{-y+3}{2}.

$$y-3-4y=-1$$

Tāpiri y ki te -4y.

$$-3y-3=-1$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-3y=2$$

Whakawehea ngā taha e rua ki te -3.

$$y=-\frac{2}{3}$$

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

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

Whakareatia -\frac{1}{2} ki te -\frac{2}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1}{3}+\frac{3}{2}$$

Tāpiri \frac{3}{2} ki te \frac{1}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{11}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{6},y=-\frac{2}{3}$$

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.

$$2x+y=3,-2x-4y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\-2&-4\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{6}\\-\frac{2}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{11}{6},y=-\frac{2}{3}$$

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.

$$2x+y=3,-2x-4y=-1$$

Kia ōrite ai a 2x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-2\times 2x-2y=-2\times 3,2\left(-2\right)x+2\left(-4\right)y=2\left(-1\right)$$

Whakarūnātia.

$$-4x-2y=-6,-4x-8y=-2$$

Me tango -4x-8y=-2 mai i -4x-2y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-2y+8y=-6+2$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -2y ki te 8y.

$$6y=-6+2$$

Tāpiri -6 ki te 2.

$$6y=-4$$

Whakawehea ngā taha e rua ki te 6.

$$y=-\frac{2}{3}$$

Whakaurua te -\frac{2}{3} mō y ki -2x-4y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-4\left(-\frac{2}{3}\right)=-1$$

Whakareatia -4 ki te -\frac{2}{3}.

$$-2x+\frac{8}{3}=-1$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$-2x=-\frac{11}{3}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{11}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{6},y=-\frac{2}{3}$$