Whakaoti i te {l}{3x-2y=13}{x+2y=-1}

$$\left\{ \begin{array} { l } { 3 x - 2 y = 13 } \\ { x + 2 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-2$$

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.

$$3x-2y=13,x+2y=-1$$

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

$$3x-2y=13$$

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

$$3x=2y+13$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y+13\right)$$

Whakareatia \frac{1}{3} ki te 2y+13.

$$x=\frac{2}{3}y+\frac{13}{3}$$

Whakakapia te \frac{2y+13}{3} mō te x ki tērā atu whārite, x+2y=-1.

$$\frac{2}{3}y+\frac{13}{3}+2y=-1$$

Tāpiri \frac{2y}{3} ki te 2y.

$$\frac{8}{3}y+\frac{13}{3}=-1$$

Me tango \frac{13}{3} mai i ngā taha e rua o te whārite.

$$\frac{8}{3}y=-\frac{16}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{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{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\left(-2\right)+\frac{13}{3}$$

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

$$x=\frac{-4+13}{3}$$

Tāpiri \frac{13}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

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

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.

$$3x-2y=13,x+2y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}3&-2\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}13\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}13\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&2\end{matrix}\right))\left(\begin{matrix}13\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-\left(-2\right)}&-\frac{-2}{3\times 2-\left(-2\right)}\\-\frac{1}{3\times 2-\left(-2\right)}&\frac{3}{3\times 2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}13\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{8}&\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}13\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 13+\frac{1}{4}\left(-1\right)\\-\frac{1}{8}\times 13+\frac{3}{8}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x-2y=13,x+2y=-1$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x-2y=13,3x+3\times 2y=3\left(-1\right)$$

Whakarūnātia.

$$3x-2y=13,3x+6y=-3$$

Me tango 3x+6y=-3 mai i 3x-2y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-2y-6y=13+3$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-6y=13+3$$

Tāpiri -2y ki te -6y.

$$-8y=13+3$$

Tāpiri 13 ki te 3.

$$-8y=16$$

Whakawehea ngā taha e rua ki te -8.

$$y=-2$$

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

$$x+2\left(-2\right)=-1$$

Whakareatia 2 ki te -2.

$$x-4=-1$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

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