Whakaoti i te {l}{x+y=17}{2x-y=11}

$$\left\{ \begin{array} { l } { x + y = 17 } \\ { 2 x - y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{28}{3} = 9\frac{1}{3} \approx 9.333333333$$

$$y = \frac{23}{3} = 7\frac{2}{3} \approx 7.666666667$$

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.

$$x+y=17,2x-y=11$$

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

$$x+y=17$$

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

$$x=-y+17$$

Whakakapia te -y+17 mō te x ki tērā atu whārite, 2x-y=11.

$$2\left(-y+17\right)-y=11$$

Whakareatia 2 ki te -y+17.

$$-2y+34-y=11$$

Tāpiri -2y ki te -y.

$$-3y+34=11$$

Me tango 34 mai i ngā taha e rua o te whārite.

$$-3y=-23$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{23}{3}$$

Whakaurua te \frac{23}{3} mō y ki x=-y+17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{23}{3}+17$$

Tāpiri 17 ki te -\frac{23}{3}.

$$x=\frac{28}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{28}{3},y=\frac{23}{3}$$

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.

$$x+y=17,2x-y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\11\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&-1\end{matrix}\right))\left(\begin{matrix}1&1\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-1\end{matrix}\right))\left(\begin{matrix}17\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-1\end{matrix}\right))\left(\begin{matrix}17\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-1\end{matrix}\right))\left(\begin{matrix}17\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-2}&-\frac{1}{-1-2}\\-\frac{2}{-1-2}&\frac{1}{-1-2}\end{matrix}\right)\left(\begin{matrix}17\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{3}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}17\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 17+\frac{1}{3}\times 11\\\frac{2}{3}\times 17-\frac{1}{3}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{28}{3}\\\frac{23}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{28}{3},y=\frac{23}{3}$$

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.

$$x+y=17,2x-y=11$$

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.

$$2x+2y=2\times 17,2x-y=11$$

Whakarūnātia.

$$2x+2y=34,2x-y=11$$

Me tango 2x-y=11 mai i 2x+2y=34 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y+y=34-11$$

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.

$$2y+y=34-11$$

Tāpiri 2y ki te y.

$$3y=34-11$$

Tāpiri 34 ki te -11.

$$3y=23$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{23}{3}$$

Whakaurua te \frac{23}{3} mō y ki 2x-y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-\frac{23}{3}=11$$

Me tāpiri \frac{23}{3} ki ngā taha e rua o te whārite.

$$2x=\frac{56}{3}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{28}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{28}{3},y=\frac{23}{3}$$