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

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

Whakaoti mō x, y

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

$$y=\frac{3}{4}=0.75$$

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.

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

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

$$3x-2y=-3$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y-3$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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

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

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

$$\frac{4}{3}y-2+4y=2$$

Tāpiri \frac{4y}{3} ki te 4y.

$$\frac{16}{3}y-2=2$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$\frac{16}{3}y=4$$

Whakawehea ngā taha e rua o te whārite ki te \frac{16}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3}{4}$$

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

$$x=\frac{2}{3}\times \frac{3}{4}-1$$

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

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

Tāpiri -1 ki te \frac{1}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{2},y=\frac{3}{4}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\2&4\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{8}\\-\frac{1}{8}&\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\left(-3\right)+\frac{1}{8}\times 2\\-\frac{1}{8}\left(-3\right)+\frac{3}{16}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{2},y=\frac{3}{4}$$

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.

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

Kia ōrite ai a 3x 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 3.

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

Whakarūnātia.

$$6x-4y=-6,6x+12y=6$$

Me tango 6x+12y=6 mai i 6x-4y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-4y-12y=-6-6$$

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.

$$-4y-12y=-6-6$$

Tāpiri -4y ki te -12y.

$$-16y=-6-6$$

Tāpiri -6 ki te -6.

$$-16y=-12$$

Whakawehea ngā taha e rua ki te -16.

$$y=\frac{3}{4}$$

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

$$2x+4\times \frac{3}{4}=2$$

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

$$2x+3=2$$

Me tango 3 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.

$$x=-\frac{1}{2},y=\frac{3}{4}$$