Whakaoti i te {c}{2x+3y=13}{x+2y=9}

$$\left. \begin{array} { c } { 2 x + 3 y = 13 } \\ { x + 2 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=5$$

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.

$$2x+3y=13,x+2y=9$$

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

$$2x+3y=13$$

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

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

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{-3y+13}{2} mō te x ki tērā atu whārite, x+2y=9.

$$-\frac{3}{2}y+\frac{13}{2}+2y=9$$

Tāpiri -\frac{3y}{2} ki te 2y.

$$\frac{1}{2}y+\frac{13}{2}=9$$

Me tango \frac{13}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{2}y=\frac{5}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{3}{2}y+\frac{13}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 5+\frac{13}{2}$$

Whakareatia -\frac{3}{2} ki te 5.

$$x=\frac{-15+13}{2}$$

Tāpiri \frac{13}{2} ki te -\frac{15}{2} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=5$$

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.

$$2x+3y=13,x+2y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}2&3\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}13\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}13\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}13\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3}&-\frac{3}{2\times 2-3}\\-\frac{1}{2\times 2-3}&\frac{2}{2\times 2-3}\end{matrix}\right)\left(\begin{matrix}13\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-3\\-1&2\end{matrix}\right)\left(\begin{matrix}13\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 13-3\times 9\\-13+2\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=5$$

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.

$$2x+3y=13,x+2y=9$$

Kia ōrite ai a 2x 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 2.

$$2x+3y=13,2x+2\times 2y=2\times 9$$

Whakarūnātia.

$$2x+3y=13,2x+4y=18$$

Me tango 2x+4y=18 mai i 2x+3y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+3y-4y=13-18$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-4y=13-18$$

Tāpiri 3y ki te -4y.

$$-y=13-18$$

Tāpiri 13 ki te -18.

$$-y=-5$$

Whakawehea ngā taha e rua ki te -1.

$$y=5$$

Whakaurua te 5 mō y ki x+2y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2\times 5=9$$

Whakareatia 2 ki te 5.

$$x+10=9$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=5$$