Whakaoti i te {l}{2x+3y=38}{-3x+2y=21}

$$\left\{ \begin{array} { l } { 2 x + 3 y = 38 } \\ { - 3 x + 2 y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=12$$

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.

$$2x+3y=38,-3x+2y=21$$

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

$$2x+3y=38$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+38$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+38\right)$$

Whakareatia \frac{1}{2} ki te -3y+38.

$$x=-\frac{3}{2}y+19$$

Whakakapia te -\frac{3y}{2}+19 mō te x ki tērā atu whārite, -3x+2y=21.

$$-3\left(-\frac{3}{2}y+19\right)+2y=21$$

Whakareatia -3 ki te -\frac{3y}{2}+19.

$$\frac{9}{2}y-57+2y=21$$

Tāpiri \frac{9y}{2} ki te 2y.

$$\frac{13}{2}y-57=21$$

Me tāpiri 57 ki ngā taha e rua o te whārite.

$$\frac{13}{2}y=78$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=12$$

Whakaurua te 12 mō y ki x=-\frac{3}{2}y+19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 12+19$$

Whakareatia -\frac{3}{2} ki te 12.

$$x=-18+19$$

Tāpiri 19 ki te -18.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=12$$

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.

$$2x+3y=38,-3x+2y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\-3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}38\\21\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\-3&2\end{matrix}\right))\left(\begin{matrix}2&3\\-3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-3&2\end{matrix}\right))\left(\begin{matrix}38\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-3&2\end{matrix}\right))\left(\begin{matrix}38\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-3&2\end{matrix}\right))\left(\begin{matrix}38\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3\left(-3\right)}&-\frac{3}{2\times 2-3\left(-3\right)}\\-\frac{-3}{2\times 2-3\left(-3\right)}&\frac{2}{2\times 2-3\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}38\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}&-\frac{3}{13}\\\frac{3}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}38\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\times 38-\frac{3}{13}\times 21\\\frac{3}{13}\times 38+\frac{2}{13}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=12$$

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.

$$2x+3y=38,-3x+2y=21$$

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.

$$-3\times 2x-3\times 3y=-3\times 38,2\left(-3\right)x+2\times 2y=2\times 21$$

Whakarūnātia.

$$-6x-9y=-114,-6x+4y=42$$

Me tango -6x+4y=42 mai i -6x-9y=-114 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x-9y-4y=-114-42$$

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.

$$-9y-4y=-114-42$$

Tāpiri -9y ki te -4y.

$$-13y=-114-42$$

Tāpiri -114 ki te -42.

$$-13y=-156$$

Whakawehea ngā taha e rua ki te -13.

$$y=12$$

Whakaurua te 12 mō y ki -3x+2y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+2\times 12=21$$

Whakareatia 2 ki te 12.

$$-3x+24=21$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=12$$