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

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

Whakaoti mō x, y

$$x=0$$

$$y=2$$

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.

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

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

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

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

$$3x=2y-4$$

Whakawehea ngā taha e rua ki te 3.

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

Whakareatia \frac{1}{3} ki te -4+2y.

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

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

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

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

$$\frac{4}{3}y-\frac{8}{3}+y=2$$

Tāpiri \frac{4y}{3} ki te y.

$$\frac{7}{3}y-\frac{8}{3}=2$$

Me tāpiri \frac{8}{3} ki ngā taha e rua o te whārite.

$$\frac{7}{3}y=\frac{14}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

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

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

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

$$x=\frac{4-4}{3}$$

Tāpiri -\frac{4}{3} ki te \frac{4}{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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{2}{7}\\-\frac{2}{7}&\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}-4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\left(-4\right)+\frac{2}{7}\times 2\\-\frac{2}{7}\left(-4\right)+\frac{3}{7}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=2$$

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.

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

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.

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

Whakarūnātia.

$$6x-4y=-8,6x+3y=6$$

Me tango 6x+3y=6 mai i 6x-4y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-4y-3y=-8-6$$

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.

$$-4y-3y=-8-6$$

Tāpiri -4y ki te -3y.

$$-7y=-8-6$$

Tāpiri -8 ki te -6.

$$-7y=-14$$

Whakawehea ngā taha e rua ki te -7.

$$y=2$$

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

$$2x+2=2$$

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

$$2x=0$$

Whakawehea ngā taha e rua ki te 2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$