Whakaoti i te {c}{2x+3y=13}{-6x+y=11}

$$\left\{ \begin{array} { c } { 2 x + 3 y = 13 } \\ { - 6 x + y = 11 } \end{array} \right.$$

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.

$$2x+3y=13,-6x+y=11$$

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, -6x+y=11.

$$-6\left(-\frac{3}{2}y+\frac{13}{2}\right)+y=11$$

Whakareatia -6 ki te \frac{-3y+13}{2}.

$$9y-39+y=11$$

Tāpiri 9y ki te y.

$$10y-39=11$$

Me tāpiri 39 ki ngā taha e rua o te whārite.

$$10y=50$$

Whakawehea ngā taha e rua ki te 10.

$$y=5$$

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.

$$2x+3y=13,-6x+y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\-6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\-6&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\-6&1\end{matrix}\right))\left(\begin{matrix}2&3\\-6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-6&1\end{matrix}\right))\left(\begin{matrix}13\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-6&1\end{matrix}\right))\left(\begin{matrix}13\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\-6&1\end{matrix}\right))\left(\begin{matrix}13\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-3\left(-6\right)}&-\frac{3}{2-3\left(-6\right)}\\-\frac{-6}{2-3\left(-6\right)}&\frac{2}{2-3\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}13\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}&-\frac{3}{20}\\\frac{3}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}13\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}\times 13-\frac{3}{20}\times 11\\\frac{3}{10}\times 13+\frac{1}{10}\times 11\end{matrix}\right)$$

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.

$$2x+3y=13,-6x+y=11$$

Kia ōrite ai a 2x me -6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-6\times 2x-6\times 3y=-6\times 13,2\left(-6\right)x+2y=2\times 11$$

Whakarūnātia.

$$-12x-18y=-78,-12x+2y=22$$

Me tango -12x+2y=22 mai i -12x-18y=-78 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x-18y-2y=-78-22$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-18y-2y=-78-22$$

Tāpiri -18y ki te -2y.

$$-20y=-78-22$$

Tāpiri -78 ki te -22.

$$-20y=-100$$

Whakawehea ngā taha e rua ki te -20.

$$y=5$$

Whakaurua te 5 mō y ki -6x+y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-6x+5=11$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-6x=6$$

Whakawehea ngā taha e rua ki te -6.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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