Whakaoti i te {l}{-x-5y=11}{2x+y=9}

$$\left. \begin{array} { l } { - x - 5 y = 11 } \\ { 2 x + y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{56}{9} = 6\frac{2}{9} \approx 6.222222222$$

$$y = -\frac{31}{9} = -3\frac{4}{9} \approx -3.444444444$$

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.

$$-x-5y=11,2x+y=9$$

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

$$-x-5y=11$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$-x=5y+11$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(5y+11\right)$$

Whakareatia -1 ki te 5y+11.

$$x=-5y-11$$

Whakakapia te -5y-11 mō te x ki tērā atu whārite, 2x+y=9.

$$2\left(-5y-11\right)+y=9$$

Whakareatia 2 ki te -5y-11.

$$-10y-22+y=9$$

Tāpiri -10y ki te y.

$$-9y-22=9$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$-9y=31$$

Whakawehea ngā taha e rua ki te -9.

$$y=-\frac{31}{9}$$

Whakaurua te -\frac{31}{9} mō y ki x=-5y-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\left(-\frac{31}{9}\right)-11$$

Whakareatia -5 ki te -\frac{31}{9}.

$$x=\frac{155}{9}-11$$

Tāpiri -11 ki te \frac{155}{9}.

$$x=\frac{56}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{56}{9},y=-\frac{31}{9}$$

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.

$$-x-5y=11,2x+y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-5\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-5\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&-5\\2&1\end{matrix}\right))\left(\begin{matrix}-1&-5\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\2&1\end{matrix}\right))\left(\begin{matrix}11\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\2&1\end{matrix}\right))\left(\begin{matrix}11\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\2&1\end{matrix}\right))\left(\begin{matrix}11\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-1-\left(-5\times 2\right)}&-\frac{-5}{-1-\left(-5\times 2\right)}\\-\frac{2}{-1-\left(-5\times 2\right)}&-\frac{1}{-1-\left(-5\times 2\right)}\end{matrix}\right)\left(\begin{matrix}11\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&\frac{5}{9}\\-\frac{2}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}11\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 11+\frac{5}{9}\times 9\\-\frac{2}{9}\times 11-\frac{1}{9}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{56}{9}\\-\frac{31}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{56}{9},y=-\frac{31}{9}$$

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.

$$-x-5y=11,2x+y=9$$

Kia ōrite ai a -x 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 -1.

$$2\left(-1\right)x+2\left(-5\right)y=2\times 11,-2x-y=-9$$

Whakarūnātia.

$$-2x-10y=22,-2x-y=-9$$

Me tango -2x-y=-9 mai i -2x-10y=22 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-10y+y=22+9$$

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.

$$-10y+y=22+9$$

Tāpiri -10y ki te y.

$$-9y=22+9$$

Tāpiri 22 ki te 9.

$$-9y=31$$

Whakawehea ngā taha e rua ki te -9.

$$y=-\frac{31}{9}$$

Whakaurua te -\frac{31}{9} mō y ki 2x+y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-\frac{31}{9}=9$$

Me tāpiri \frac{31}{9} ki ngā taha e rua o te whārite.

$$2x=\frac{112}{9}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{56}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{56}{9},y=-\frac{31}{9}$$